from functools import partial
from typing import Union, Callable, Optional, Any, Sequence
from jax.lax import stop_gradient
import brainpy.math as bm
from brainpy._src.context import share
from brainpy._src.dyn._docs import ref_doc, lif_doc, pneu_doc, dpneu_doc, ltc_doc, if_doc
from brainpy._src.dyn.neurons.base import GradNeuDyn
from brainpy._src.initialize import ZeroInit, OneInit, noise as init_noise
from brainpy._src.integrators import odeint, sdeint, JointEq
from brainpy.check import is_initializer
from brainpy.types import Shape, ArrayType, Sharding
__all__ = [
'IF',
'IFLTC',
'Lif',
'LifLTC',
'LifRef',
'LifRefLTC',
'ExpIF',
'ExpIFLTC',
'ExpIFRef',
'ExpIFRefLTC',
'AdExIF',
'AdExIFLTC',
'AdExIFRef',
'AdExIFRefLTC',
'QuaIF',
'QuaIFLTC',
'QuaIFRef',
'QuaIFRefLTC',
'AdQuaIF',
'AdQuaIFLTC',
'AdQuaIFRef',
'AdQuaIFRefLTC',
'Gif',
'GifLTC',
'GifRef',
'GifRefLTC',
'Izhikevich',
'IzhikevichLTC',
'IzhikevichRef',
'IzhikevichRefLTC',
]
class IFLTC(GradNeuDyn):
r"""Leaky Integrator Model %s.
**Model Descriptions**
This class implements a leaky integrator model, in which its dynamics is
given by:
.. math::
\tau \frac{dV}{dt} = - (V(t) - V_{rest}) + RI(t)
where :math:`V` is the membrane potential, :math:`V_{rest}` is the resting
membrane potential, :math:`\tau` is the time constant, and :math:`R` is the
resistance.
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = 0.,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.tau = self.init_param(tau)
self.R = self.init_param(R)
# initializers
self._V_initializer = is_initializer(V_initializer)
# integral
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def derivative(self, V, t, I):
I = self.sum_current_inputs(V, init=I)
return (-V + self.V_rest + self.R * I) / self.tau
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
self.V.value = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
return self.V.value
def return_info(self):
return self.V
class IF(IFLTC):
def derivative(self, V, t, I):
return (-V + self.V_rest + self.R * I) / self.tau
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
IF.__doc__ = IFLTC.__doc__ % ('', if_doc, pneu_doc, dpneu_doc)
IFLTC.__doc__ = IFLTC.__doc__ % (ltc_doc, if_doc, pneu_doc, dpneu_doc)
[docs]
class LifLTC(GradNeuDyn):
r"""Leaky integrate-and-fire neuron model with liquid time-constant.
The formal equations of a LIF model [1]_ is given by:
.. math::
\tau \frac{dV}{dt} = - (V(t) - V_{rest}) + RI(t) \\
\text{after} \quad V(t) \gt V_{th}, V(t) = V_{reset}
where :math:`V` is the membrane potential, :math:`V_{rest}` is the resting
membrane potential, :math:`V_{reset}` is the reset membrane potential,
:math:`V_{th}` is the spike threshold, :math:`\tau` is the time constant,
and :math:`I` is the time-variant synaptic inputs.
.. [1] Abbott, Larry F. "Lapicque’s introduction of the integrate-and-fire model
neuron (1907)." Brain research bulletin 50, no. 5-6 (1999): 303-304.
**Examples**
There is an example usage: mustang u r lvd by the blonde boy
.. code-block:: python
import brainpy as bp
lif = bp.dyn.LifLTC(1)
# raise input current from 4 mA to 40 mA
inputs = bp.inputs.ramp_input(4, 40, 700, 100, 600,)
runner = bp.DSRunner(lif, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = 0.,
V_reset: Union[float, ArrayType, Callable] = -5.,
V_th: Union[float, ArrayType, Callable] = 20.,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# noise
noise: Optional[Union[float, ArrayType, Callable]] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th = self.offset_scaling(self.init_param(V_th))
self.tau = self.init_param(tau)
self.R = self.init_param(R)
# initializers
self._V_initializer = is_initializer(V_initializer)
# noise
self.noise = init_noise(noise, self.varshape)
# integral
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def derivative(self, V, t, I):
I = self.sum_current_inputs(V, init=I)
return (-V + self.V_rest + self.R * I) / self.tau
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
self.V.value = V
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class Lif(LifLTC):
r"""Leaky integrate-and-fire neuron model.
The formal equations of a LIF model [1]_ is given by:
.. math::
\tau \frac{dV}{dt} = - (V(t) - V_{rest}) + RI(t) \\
\text{after} \quad V(t) \gt V_{th}, V(t) = V_{reset}
where :math:`V` is the membrane potential, :math:`V_{rest}` is the resting
membrane potential, :math:`V_{reset}` is the reset membrane potential,
:math:`V_{th}` is the spike threshold, :math:`\tau` is the time constant,
and :math:`I` is the time-variant synaptic inputs.
.. [1] Abbott, Larry F. "Lapicque’s introduction of the integrate-and-fire model
neuron (1907)." Brain research bulletin 50, no. 5-6 (1999): 303-304.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
lif = bp.dyn.Lif(1)
# raise input current from 4 mA to 40 mA
inputs = bp.inputs.ramp_input(4, 40, 700, 100, 600,)
runner = bp.DSRunner(lif, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
Args:
%s
%s
%s
"""
def derivative(self, V, t, I):
return (-V + self.V_rest + self.R * I) / self.tau
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
Lif.__doc__ = Lif.__doc__ % (lif_doc, pneu_doc, dpneu_doc)
LifLTC.__doc__ = LifLTC.__doc__ % (lif_doc, pneu_doc, dpneu_doc)
[docs]
class LifRefLTC(LifLTC):
r"""Leaky integrate-and-fire neuron model with liquid time-constant which has refractory periods .
The formal equations of a LIF model [1]_ is given by:
.. math::
\tau \frac{dV}{dt} = - (V(t) - V_{rest}) + RI(t) \\
\text{after} \quad V(t) \gt V_{th}, V(t) = V_{reset} \quad
\text{last} \quad \tau_{ref} \quad \text{ms}
where :math:`V` is the membrane potential, :math:`V_{rest}` is the resting
membrane potential, :math:`V_{reset}` is the reset membrane potential,
:math:`V_{th}` is the spike threshold, :math:`\tau` is the time constant,
:math:`\tau_{ref}` is the refractory time period,
and :math:`I` is the time-variant synaptic inputs.
.. [1] Abbott, Larry F. "Lapicque’s introduction of the integrate-and-fire model
neuron (1907)." Brain research bulletin 50, no. 5-6 (1999): 303-304.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.LifRefLTC(1, )
# example for section input
inputs = bp.inputs.section_input([0., 21., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
Args:
%s
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
detach_spk: bool = False,
spk_reset: str = 'soft',
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = 0.,
V_reset: Union[float, ArrayType, Callable] = -5.,
V_th: Union[float, ArrayType, Callable] = 20.,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Optional[Union[float, ArrayType, Callable]] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th=V_th,
R=R,
tau=tau,
V_initializer=V_initializer,
noise=noise,
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e7)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class LifRef(LifRefLTC):
r"""Leaky integrate-and-fire neuron model %s which has refractory periods.
The formal equations of a LIF model [1]_ is given by:
.. math::
\tau \frac{dV}{dt} = - (V(t) - V_{rest}) + RI(t) \\
\text{after} \quad V(t) \gt V_{th}, V(t) = V_{reset} \quad
\text{last} \quad \tau_{ref} \quad \text{ms}
where :math:`V` is the membrane potential, :math:`V_{rest}` is the resting
membrane potential, :math:`V_{reset}` is the reset membrane potential,
:math:`V_{th}` is the spike threshold, :math:`\tau` is the time constant,
:math:`\tau_{ref}` is the refractory time period,
and :math:`I` is the time-variant synaptic inputs.
.. [1] Abbott, Larry F. "Lapicque’s introduction of the integrate-and-fire model
neuron (1907)." Brain research bulletin 50, no. 5-6 (1999): 303-304.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.LifRef(1, )
# example for section input
inputs = bp.inputs.section_input([0., 21., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
Args:
%s
%s
%s
%s
"""
def derivative(self, V, t, I):
return (-V + self.V_rest + self.R * I) / self.tau
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
LifRef.__doc__ = LifRefLTC.__doc__ % (lif_doc, pneu_doc, dpneu_doc, ref_doc)
LifRefLTC.__doc__ = LifRefLTC.__doc__ % (lif_doc, pneu_doc, dpneu_doc, ref_doc)
[docs]
class ExpIFLTC(GradNeuDyn):
r"""Exponential integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
In the exponential integrate-and-fire model [1]_, the differential
equation for the membrane potential is given by
.. math::
\tau\frac{d V}{d t}= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} + RI(t), \\
\text{after} \, V(t) \gt V_{th}, V(t) = V_{reset} \, \text{last} \, \tau_{ref} \, \text{ms}
This equation has an exponential nonlinearity with "sharpness" parameter :math:`\Delta_{T}`
and "threshold" :math:`\vartheta_{rh}`.
The moment when the membrane potential reaches the numerical threshold :math:`V_{th}`
defines the firing time :math:`t^{(f)}`. After firing, the membrane potential is reset to
:math:`V_{rest}` and integration restarts at time :math:`t^{(f)}+\tau_{\rm ref}`,
where :math:`\tau_{\rm ref}` is an absolute refractory time.
If the numerical threshold is chosen sufficiently high, :math:`V_{th}\gg v+\Delta_T`,
its exact value does not play any role. The reason is that the upswing of the action
potential for :math:`v\gg v +\Delta_{T}` is so rapid, that it goes to infinity in
an incredibly short time. The threshold :math:`V_{th}` is introduced mainly for numerical
convenience. For a formal mathematical analysis of the model, the threshold can be pushed
to infinity.
The model was first introduced by Nicolas Fourcaud-Trocmé, David Hansel, Carl van Vreeswijk
and Nicolas Brunel [1]_. The exponential nonlinearity was later confirmed by Badel et al. [3]_.
It is one of the prominent examples of a precise theoretical prediction in computational
neuroscience that was later confirmed by experimental neuroscience.
Two important remarks:
- (i) The right-hand side of the above equation contains a nonlinearity
that can be directly extracted from experimental data [3]_. In this sense the exponential
nonlinearity is not an arbitrary choice but directly supported by experimental evidence.
- (ii) Even though it is a nonlinear model, it is simple enough to calculate the firing
rate for constant input, and the linear response to fluctuations, even in the presence
of input noise [4]_.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014).
Neuronal dynamics: From single neurons to networks and models
of cognition. Cambridge University Press.
.. [3] Badel, Laurent, Sandrine Lefort, Romain Brette, Carl CH Petersen,
Wulfram Gerstner, and Magnus JE Richardson. "Dynamic IV curves
are reliable predictors of naturalistic pyramidal-neuron voltage
traces." Journal of Neurophysiology 99, no. 2 (2008): 656-666.
.. [4] Richardson, Magnus JE. "Firing-rate response of linear and nonlinear
integrate-and-fire neurons to modulated current-based and
conductance-based synaptic drive." Physical Review E 76, no. 2 (2007): 021919.
.. [5] https://en.wikipedia.org/wiki/Exponential_integrate-and-fire
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.ExpIFLTC(1, )
# example for section input
inputs = bp.inputs.section_input([0., 5., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
**Model Parameters**
============= ============== ======== ===================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ---------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
R 1 \ Membrane resistance.
tau 10 \ Membrane time constant. Compute by R * C.
tau_ref 1.7 \ Refractory period length.
============= ============== ======== ===================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -55.,
V_T: Union[float, ArrayType, Callable] = -59.9,
delta_T: Union[float, ArrayType, Callable] = 3.48,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th = self.offset_scaling(self.init_param(V_th))
self.V_T = self.offset_scaling(self.init_param(V_T))
self.delta_T = self.std_scaling(self.init_param(delta_T))
self.tau = self.init_param(tau)
self.R = self.init_param(R)
# initializers
self._V_initializer = is_initializer(V_initializer)
# noise
self.noise = init_noise(noise, self.varshape)
# integral
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def derivative(self, V, t, I):
I = self.sum_current_inputs(V, init=I)
exp_v = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dvdt = (- (V - self.V_rest) + exp_v + self.R * I) / self.tau
return dvdt
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
self.V.value = V
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class ExpIF(ExpIFLTC):
r"""Exponential integrate-and-fire neuron model.
**Model Descriptions**
In the exponential integrate-and-fire model [1]_, the differential
equation for the membrane potential is given by
.. math::
\tau\frac{d V}{d t}= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} + RI(t), \\
\text{after} \, V(t) \gt V_{th}, V(t) = V_{reset} \, \text{last} \, \tau_{ref} \, \text{ms}
This equation has an exponential nonlinearity with "sharpness" parameter :math:`\Delta_{T}`
and "threshold" :math:`\vartheta_{rh}`.
The moment when the membrane potential reaches the numerical threshold :math:`V_{th}`
defines the firing time :math:`t^{(f)}`. After firing, the membrane potential is reset to
:math:`V_{rest}` and integration restarts at time :math:`t^{(f)}+\tau_{\rm ref}`,
where :math:`\tau_{\rm ref}` is an absolute refractory time.
If the numerical threshold is chosen sufficiently high, :math:`V_{th}\gg v+\Delta_T`,
its exact value does not play any role. The reason is that the upswing of the action
potential for :math:`v\gg v +\Delta_{T}` is so rapid, that it goes to infinity in
an incredibly short time. The threshold :math:`V_{th}` is introduced mainly for numerical
convenience. For a formal mathematical analysis of the model, the threshold can be pushed
to infinity.
The model was first introduced by Nicolas Fourcaud-Trocmé, David Hansel, Carl van Vreeswijk
and Nicolas Brunel [1]_. The exponential nonlinearity was later confirmed by Badel et al. [3]_.
It is one of the prominent examples of a precise theoretical prediction in computational
neuroscience that was later confirmed by experimental neuroscience.
Two important remarks:
- (i) The right-hand side of the above equation contains a nonlinearity
that can be directly extracted from experimental data [3]_. In this sense the exponential
nonlinearity is not an arbitrary choice but directly supported by experimental evidence.
- (ii) Even though it is a nonlinear model, it is simple enough to calculate the firing
rate for constant input, and the linear response to fluctuations, even in the presence
of input noise [4]_.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014).
Neuronal dynamics: From single neurons to networks and models
of cognition. Cambridge University Press.
.. [3] Badel, Laurent, Sandrine Lefort, Romain Brette, Carl CH Petersen,
Wulfram Gerstner, and Magnus JE Richardson. "Dynamic IV curves
are reliable predictors of naturalistic pyramidal-neuron voltage
traces." Journal of Neurophysiology 99, no. 2 (2008): 656-666.
.. [4] Richardson, Magnus JE. "Firing-rate response of linear and nonlinear
integrate-and-fire neurons to modulated current-based and
conductance-based synaptic drive." Physical Review E 76, no. 2 (2007): 021919.
.. [5] https://en.wikipedia.org/wiki/Exponential_integrate-and-fire
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.ExpIF(1, )
# example for section input
inputs = bp.inputs.section_input([0., 5., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
**Model Parameters**
============= ============== ======== ===================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ---------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
R 1 \ Membrane resistance.
tau 10 \ Membrane time constant. Compute by R * C.
tau_ref 1.7 \ Refractory period length.
============= ============== ======== ===================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
"""
def derivative(self, V, t, I):
exp_v = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dvdt = (- (V - self.V_rest) + exp_v + self.R * I) / self.tau
return dvdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class ExpIFRefLTC(ExpIFLTC):
r"""Exponential integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
In the exponential integrate-and-fire model [1]_, the differential
equation for the membrane potential is given by
.. math::
\tau\frac{d V}{d t}= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} + RI(t), \\
\text{after} \, V(t) \gt V_{th}, V(t) = V_{reset} \, \text{last} \, \tau_{ref} \, \text{ms}
This equation has an exponential nonlinearity with "sharpness" parameter :math:`\Delta_{T}`
and "threshold" :math:`\vartheta_{rh}`.
The moment when the membrane potential reaches the numerical threshold :math:`V_{th}`
defines the firing time :math:`t^{(f)}`. After firing, the membrane potential is reset to
:math:`V_{rest}` and integration restarts at time :math:`t^{(f)}+\tau_{\rm ref}`,
where :math:`\tau_{\rm ref}` is an absolute refractory time.
If the numerical threshold is chosen sufficiently high, :math:`V_{th}\gg v+\Delta_T`,
its exact value does not play any role. The reason is that the upswing of the action
potential for :math:`v\gg v +\Delta_{T}` is so rapid, that it goes to infinity in
an incredibly short time. The threshold :math:`V_{th}` is introduced mainly for numerical
convenience. For a formal mathematical analysis of the model, the threshold can be pushed
to infinity.
The model was first introduced by Nicolas Fourcaud-Trocmé, David Hansel, Carl van Vreeswijk
and Nicolas Brunel [1]_. The exponential nonlinearity was later confirmed by Badel et al. [3]_.
It is one of the prominent examples of a precise theoretical prediction in computational
neuroscience that was later confirmed by experimental neuroscience.
Two important remarks:
- (i) The right-hand side of the above equation contains a nonlinearity
that can be directly extracted from experimental data [3]_. In this sense the exponential
nonlinearity is not an arbitrary choice but directly supported by experimental evidence.
- (ii) Even though it is a nonlinear model, it is simple enough to calculate the firing
rate for constant input, and the linear response to fluctuations, even in the presence
of input noise [4]_.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014).
Neuronal dynamics: From single neurons to networks and models
of cognition. Cambridge University Press.
.. [3] Badel, Laurent, Sandrine Lefort, Romain Brette, Carl CH Petersen,
Wulfram Gerstner, and Magnus JE Richardson. "Dynamic IV curves
are reliable predictors of naturalistic pyramidal-neuron voltage
traces." Journal of Neurophysiology 99, no. 2 (2008): 656-666.
.. [4] Richardson, Magnus JE. "Firing-rate response of linear and nonlinear
integrate-and-fire neurons to modulated current-based and
conductance-based synaptic drive." Physical Review E 76, no. 2 (2007): 021919.
.. [5] https://en.wikipedia.org/wiki/Exponential_integrate-and-fire
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.ExpIFRefLTC(1, )
# example for section input
inputs = bp.inputs.section_input([0., 5., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
**Model Parameters**
============= ============== ======== ===================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ---------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
R 1 \ Membrane resistance.
tau 10 \ Membrane time constant. Compute by R * C.
tau_ref 1.7 \ Refractory period length.
============= ============== ======== ===================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
detach_spk: bool = False,
spk_reset: str = 'soft',
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -55.,
V_T: Union[float, ArrayType, Callable] = -59.9,
delta_T: Union[float, ArrayType, Callable] = 3.48,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th=V_th,
V_T=V_T,
delta_T=delta_T,
R=R,
tau=tau,
V_initializer=V_initializer,
noise=noise,
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
# integral
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e7)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class ExpIFRef(ExpIFRefLTC):
r"""Exponential integrate-and-fire neuron model .
**Model Descriptions**
In the exponential integrate-and-fire model [1]_, the differential
equation for the membrane potential is given by
.. math::
\tau\frac{d V}{d t}= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} + RI(t), \\
\text{after} \, V(t) \gt V_{th}, V(t) = V_{reset} \, \text{last} \, \tau_{ref} \, \text{ms}
This equation has an exponential nonlinearity with "sharpness" parameter :math:`\Delta_{T}`
and "threshold" :math:`\vartheta_{rh}`.
The moment when the membrane potential reaches the numerical threshold :math:`V_{th}`
defines the firing time :math:`t^{(f)}`. After firing, the membrane potential is reset to
:math:`V_{rest}` and integration restarts at time :math:`t^{(f)}+\tau_{\rm ref}`,
where :math:`\tau_{\rm ref}` is an absolute refractory time.
If the numerical threshold is chosen sufficiently high, :math:`V_{th}\gg v+\Delta_T`,
its exact value does not play any role. The reason is that the upswing of the action
potential for :math:`v\gg v +\Delta_{T}` is so rapid, that it goes to infinity in
an incredibly short time. The threshold :math:`V_{th}` is introduced mainly for numerical
convenience. For a formal mathematical analysis of the model, the threshold can be pushed
to infinity.
The model was first introduced by Nicolas Fourcaud-Trocmé, David Hansel, Carl van Vreeswijk
and Nicolas Brunel [1]_. The exponential nonlinearity was later confirmed by Badel et al. [3]_.
It is one of the prominent examples of a precise theoretical prediction in computational
neuroscience that was later confirmed by experimental neuroscience.
Two important remarks:
- (i) The right-hand side of the above equation contains a nonlinearity
that can be directly extracted from experimental data [3]_. In this sense the exponential
nonlinearity is not an arbitrary choice but directly supported by experimental evidence.
- (ii) Even though it is a nonlinear model, it is simple enough to calculate the firing
rate for constant input, and the linear response to fluctuations, even in the presence
of input noise [4]_.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014).
Neuronal dynamics: From single neurons to networks and models
of cognition. Cambridge University Press.
.. [3] Badel, Laurent, Sandrine Lefort, Romain Brette, Carl CH Petersen,
Wulfram Gerstner, and Magnus JE Richardson. "Dynamic IV curves
are reliable predictors of naturalistic pyramidal-neuron voltage
traces." Journal of Neurophysiology 99, no. 2 (2008): 656-666.
.. [4] Richardson, Magnus JE. "Firing-rate response of linear and nonlinear
integrate-and-fire neurons to modulated current-based and
conductance-based synaptic drive." Physical Review E 76, no. 2 (2007): 021919.
.. [5] https://en.wikipedia.org/wiki/Exponential_integrate-and-fire
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.ExpIFRef(1, )
# example for section input
inputs = bp.inputs.section_input([0., 5., 0.], [100., 300., 100.])
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], show=True)
**Model Parameters**
============= ============== ======== ===================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ---------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
R 1 \ Membrane resistance.
tau 10 \ Membrane time constant. Compute by R * C.
tau_ref 1.7 \ Refractory period length.
============= ============== ======== ===================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def derivative(self, V, t, I):
exp_v = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dvdt = (- (V - self.V_rest) + exp_v + self.R * I) / self.tau
return dvdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
ExpIF.__doc__ = ExpIF.__doc__ % (pneu_doc, dpneu_doc)
ExpIFRefLTC.__doc__ = ExpIFRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
ExpIFRef.__doc__ = ExpIFRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
ExpIFLTC.__doc__ = ExpIFLTC.__doc__ % ()
[docs]
class AdExIFLTC(GradNeuDyn):
r"""Adaptive exponential integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
The **adaptive exponential integrate-and-fire model**, also called AdEx, is a
spiking neuron model with two variables [1]_ [2]_.
.. math::
\begin{aligned}
\tau_m\frac{d V}{d t} &= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} - Rw + RI(t), \\
\tau_w \frac{d w}{d t} &=a(V-V_{rest}) - w
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
The first equation describes the dynamics of the membrane potential and includes
an activation term with an exponential voltage dependence. Voltage is coupled to
a second equation which describes adaptation. Both variables are reset if an action
potential has been triggered. The combination of adaptation and exponential voltage
dependence gives rise to the name Adaptive Exponential Integrate-and-Fire model.
The adaptive exponential integrate-and-fire model is capable of describing known
neuronal firing patterns, e.g., adapting, bursting, delayed spike initiation,
initial bursting, fast spiking, and regular spiking.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model
**Examples**
An example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdExIFLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Examples for different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Gerstner_2005_AdExIF_model.html>`_
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
a 1 \ The sensitivity of the recovery variable :math:`u` to the sub-threshold fluctuations of the membrane potential :math:`v`
b 1 \ The increment of :math:`w` produced by a spike.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_w 30 ms Time constant of the adaptation current.
tau_ref 0. ms Refractory time.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -55.,
V_T: Union[float, ArrayType, Callable] = -59.9,
delta_T: Union[float, ArrayType, Callable] = 3.48,
a: Union[float, ArrayType, Callable] = 1.,
b: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
tau_w: Union[float, ArrayType, Callable] = 30.,
R: Union[float, ArrayType, Callable] = 1.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
w_initializer: Union[Callable, ArrayType] = ZeroInit(),
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th = self.offset_scaling(self.init_param(V_th))
self.V_T = self.offset_scaling(self.init_param(V_T))
self.a = self.init_param(a)
self.b = self.std_scaling(self.init_param(b))
self.R = self.init_param(R)
self.delta_T = self.std_scaling(self.init_param(delta_T))
self.tau = self.init_param(tau)
self.tau_w = self.init_param(tau_w)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._w_initializer = is_initializer(w_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def dV(self, V, t, w, I):
I = self.sum_current_inputs(V, init=I)
exp = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dVdt = (- V + self.V_rest + exp - self.R * w + self.R * I) / self.tau
return dVdt
def dw(self, w, t, V):
dwdt = (self.a * (V - self.V_rest) - w) / self.tau_w
return dwdt
@property
def derivative(self):
return JointEq([self.dV, self.dw])
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.w = self.std_scaling(self.init_variable(self._w_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, w = self.integral(self.V.value, self.w.value, t, x, dt)
V += self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
w += self.b * spike
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
w = bm.where(spike, w + self.b, w)
self.V.value = V
self.w.value = w
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class AdExIF(AdExIFLTC):
r"""Adaptive exponential integrate-and-fire neuron model.
**Model Descriptions**
The **adaptive exponential integrate-and-fire model**, also called AdEx, is a
spiking neuron model with two variables [1]_ [2]_.
.. math::
\begin{aligned}
\tau_m\frac{d V}{d t} &= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} - Rw + RI(t), \\
\tau_w \frac{d w}{d t} &=a(V-V_{rest}) - w
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
The first equation describes the dynamics of the membrane potential and includes
an activation term with an exponential voltage dependence. Voltage is coupled to
a second equation which describes adaptation. Both variables are reset if an action
potential has been triggered. The combination of adaptation and exponential voltage
dependence gives rise to the name Adaptive Exponential Integrate-and-Fire model.
The adaptive exponential integrate-and-fire model is capable of describing known
neuronal firing patterns, e.g., adapting, bursting, delayed spike initiation,
initial bursting, fast spiking, and regular spiking.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model
**Examples**
An example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdExIF(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Examples for different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Gerstner_2005_AdExIF_model.html>`_
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
a 1 \ The sensitivity of the recovery variable :math:`u` to the sub-threshold fluctuations of the membrane potential :math:`v`
b 1 \ The increment of :math:`w` produced by a spike.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_w 30 ms Time constant of the adaptation current.
tau_ref 0. ms Refractory time.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
"""
def dV(self, V, t, w, I):
exp = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dVdt = (- V + self.V_rest + exp - self.R * w + self.R * I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class AdExIFRefLTC(AdExIFLTC):
r"""Adaptive exponential integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
The **adaptive exponential integrate-and-fire model**, also called AdEx, is a
spiking neuron model with two variables [1]_ [2]_.
.. math::
\begin{aligned}
\tau_m\frac{d V}{d t} &= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} - Rw + RI(t), \\
\tau_w \frac{d w}{d t} &=a(V-V_{rest}) - w
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
The first equation describes the dynamics of the membrane potential and includes
an activation term with an exponential voltage dependence. Voltage is coupled to
a second equation which describes adaptation. Both variables are reset if an action
potential has been triggered. The combination of adaptation and exponential voltage
dependence gives rise to the name Adaptive Exponential Integrate-and-Fire model.
The adaptive exponential integrate-and-fire model is capable of describing known
neuronal firing patterns, e.g., adapting, bursting, delayed spike initiation,
initial bursting, fast spiking, and regular spiking.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model
**Examples**
An example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdExIFRefLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Examples for different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Gerstner_2005_AdExIF_model.html>`_
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
a 1 \ The sensitivity of the recovery variable :math:`u` to the sub-threshold fluctuations of the membrane potential :math:`v`
b 1 \ The increment of :math:`w` produced by a spike.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_w 30 ms Time constant of the adaptation current.
tau_ref 0. ms Refractory time.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -55.,
V_T: Union[float, ArrayType, Callable] = -59.9,
delta_T: Union[float, ArrayType, Callable] = 3.48,
a: Union[float, ArrayType, Callable] = 1.,
b: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
tau_w: Union[float, ArrayType, Callable] = 30.,
R: Union[float, ArrayType, Callable] = 1.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
w_initializer: Union[Callable, ArrayType] = ZeroInit(),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th=V_th,
V_T=V_T,
delta_T=delta_T,
a=a,
b=b,
R=R,
tau=tau,
tau_w=tau_w,
V_initializer=V_initializer,
w_initializer=w_initializer
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._w_initializer = is_initializer(w_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e8)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, w = self.integral(self.V.value, self.w.value, t, x, dt)
V += self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
w += self.b * spike_no_grad
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
w = bm.where(spike, w + self.b, w)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.w.value = w
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class AdExIFRef(AdExIFRefLTC):
r"""Adaptive exponential integrate-and-fire neuron model.
**Model Descriptions**
The **adaptive exponential integrate-and-fire model**, also called AdEx, is a
spiking neuron model with two variables [1]_ [2]_.
.. math::
\begin{aligned}
\tau_m\frac{d V}{d t} &= - (V-V_{rest}) + \Delta_T e^{\frac{V-V_T}{\Delta_T}} - Rw + RI(t), \\
\tau_w \frac{d w}{d t} &=a(V-V_{rest}) - w
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
The first equation describes the dynamics of the membrane potential and includes
an activation term with an exponential voltage dependence. Voltage is coupled to
a second equation which describes adaptation. Both variables are reset if an action
potential has been triggered. The combination of adaptation and exponential voltage
dependence gives rise to the name Adaptive Exponential Integrate-and-Fire model.
The adaptive exponential integrate-and-fire model is capable of describing known
neuronal firing patterns, e.g., adapting, bursting, delayed spike initiation,
initial bursting, fast spiking, and regular spiking.
**References**
.. [1] Fourcaud-Trocmé, Nicolas, et al. "How spike generation
mechanisms determine the neuronal response to fluctuating
inputs." Journal of Neuroscience 23.37 (2003): 11628-11640.
.. [2] http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model
**Examples**
Here is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdExIFRef(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Examples for different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Gerstner_2005_AdExIF_model.html>`_
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_T -59.9 mV Threshold potential of generating action potential.
delta_T 3.48 \ Spike slope factor.
a 1 \ The sensitivity of the recovery variable :math:`u` to the sub-threshold fluctuations of the membrane potential :math:`v`
b 1 \ The increment of :math:`w` produced by a spike.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_w 30 ms Time constant of the adaptation current.
tau_ref 0. ms Refractory time.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def dV(self, V, t, w, I):
exp = self.delta_T * bm.exp((V - self.V_T) / self.delta_T)
dVdt = (- V + self.V_rest + exp - self.R * w + self.R * I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
AdExIF.__doc__ = AdExIF.__doc__ % (pneu_doc, dpneu_doc)
AdExIFRefLTC.__doc__ = AdExIFRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
AdExIFRef.__doc__ = AdExIFRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
AdExIFLTC.__doc__ = AdExIFLTC.__doc__ % ()
[docs]
class QuaIFLTC(GradNeuDyn):
r"""Quadratic Integrate-and-Fire neuron model with liquid time-constant.
**Model Descriptions**
In contrast to physiologically accurate but computationally expensive
neuron models like the Hodgkin–Huxley model, the QIF model [1]_ seeks only
to produce **action potential-like patterns** and ignores subtleties
like gating variables, which play an important role in generating action
potentials in a real neuron. However, the QIF model is incredibly easy
to implement and compute, and relatively straightforward to study and
understand, thus has found ubiquitous use in computational neuroscience.
.. math::
\tau \frac{d V}{d t}=c(V-V_{rest})(V-V_c) + RI(t)
where the parameters are taken to be :math:`c` =0.07, and :math:`V_c = -50 mV` (Latham et al., 2000).
**References**
.. [1] P. E. Latham, B.J. Richmond, P. Nelson and S. Nirenberg
(2000) Intrinsic dynamics in neuronal networks. I. Theory.
J. Neurophysiology 83, pp. 808–827.
**Examples**
Here is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.QuaIFLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger than V_rest.
c .07 \ Coefficient describes membrane potential update. Larger than 0.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_ref 0 ms Refractory period length.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -30.,
V_c: Union[float, ArrayType, Callable] = -50.0,
c: Union[float, ArrayType, Callable] = 0.07,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th = self.offset_scaling(self.init_param(V_th))
self.V_c = self.offset_scaling(self.init_param(V_c))
self.c = self.inv_scaling(self.init_param(c))
self.R = self.init_param(R)
self.tau = self.init_param(tau)
# initializers
self._V_initializer = is_initializer(V_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=1)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def derivative(self, V, t, I):
I = self.sum_current_inputs(V, init=I)
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) + self.R * I) / self.tau
return dVdt
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
self.V.value = V
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class QuaIF(QuaIFLTC):
r"""Quadratic Integrate-and-Fire neuron model.
**Model Descriptions**
In contrast to physiologically accurate but computationally expensive
neuron models like the Hodgkin–Huxley model, the QIF model [1]_ seeks only
to produce **action potential-like patterns** and ignores subtleties
like gating variables, which play an important role in generating action
potentials in a real neuron. However, the QIF model is incredibly easy
to implement and compute, and relatively straightforward to study and
understand, thus has found ubiquitous use in computational neuroscience.
.. math::
\tau \frac{d V}{d t}=c(V-V_{rest})(V-V_c) + RI(t)
where the parameters are taken to be :math:`c` =0.07, and :math:`V_c = -50 mV` (Latham et al., 2000).
**References**
.. [1] P. E. Latham, B.J. Richmond, P. Nelson and S. Nirenberg
(2000) Intrinsic dynamics in neuronal networks. I. Theory.
J. Neurophysiology 83, pp. 808–827.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.QuaIF(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger than V_rest.
c .07 \ Coefficient describes membrane potential update. Larger than 0.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_ref 0 ms Refractory period length.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
"""
def derivative(self, V, t, I):
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) + self.R * I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class QuaIFRefLTC(QuaIFLTC):
r"""Quadratic Integrate-and-Fire neuron model with liquid time-constant.
**Model Descriptions**
In contrast to physiologically accurate but computationally expensive
neuron models like the Hodgkin–Huxley model, the QIF model [1]_ seeks only
to produce **action potential-like patterns** and ignores subtleties
like gating variables, which play an important role in generating action
potentials in a real neuron. However, the QIF model is incredibly easy
to implement and compute, and relatively straightforward to study and
understand, thus has found ubiquitous use in computational neuroscience.
.. math::
\tau \frac{d V}{d t}=c(V-V_{rest})(V-V_c) + RI(t)
where the parameters are taken to be :math:`c` =0.07, and :math:`V_c = -50 mV` (Latham et al., 2000).
**References**
.. [1] P. E. Latham, B.J. Richmond, P. Nelson and S. Nirenberg
(2000) Intrinsic dynamics in neuronal networks. I. Theory.
J. Neurophysiology 83, pp. 808–827.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.QuaIFRefLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger than V_rest.
c .07 \ Coefficient describes membrane potential update. Larger than 0.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_ref 0 ms Refractory period length.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -30.,
V_c: Union[float, ArrayType, Callable] = -50.0,
c: Union[float, ArrayType, Callable] = 0.07,
R: Union[float, ArrayType, Callable] = 1.,
tau: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th=V_th,
V_c=V_c,
c=c,
R=R,
tau=tau,
V_initializer=V_initializer,
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=1)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e7)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V = self.integral(self.V.value, t, x, dt) + self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class QuaIFRef(QuaIFRefLTC):
r"""Quadratic Integrate-and-Fire neuron model.
**Model Descriptions**
In contrast to physiologically accurate but computationally expensive
neuron models like the Hodgkin–Huxley model, the QIF model [1]_ seeks only
to produce **action potential-like patterns** and ignores subtleties
like gating variables, which play an important role in generating action
potentials in a real neuron. However, the QIF model is incredibly easy
to implement and compute, and relatively straightforward to study and
understand, thus has found ubiquitous use in computational neuroscience.
.. math::
\tau \frac{d V}{d t}=c(V-V_{rest})(V-V_c) + RI(t)
where the parameters are taken to be :math:`c` =0.07, and :math:`V_c = -50 mV` (Latham et al., 2000).
**References**
.. [1] P. E. Latham, B.J. Richmond, P. Nelson and S. Nirenberg
(2000) Intrinsic dynamics in neuronal networks. I. Theory.
J. Neurophysiology 83, pp. 808–827.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.QuaIFRef(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== ========================================================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- ------------------------------------------------------------------------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger than V_rest.
c .07 \ Coefficient describes membrane potential update. Larger than 0.
R 1 \ Membrane resistance.
tau 10 ms Membrane time constant. Compute by R * C.
tau_ref 0 ms Refractory period length.
============= ============== ======== ========================================================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V 0 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def derivative(self, V, t, I):
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) + self.R * I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
QuaIF.__doc__ = QuaIF.__doc__ % (pneu_doc, dpneu_doc)
QuaIFRefLTC.__doc__ = QuaIFRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
QuaIFRef.__doc__ = QuaIFRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
QuaIFLTC.__doc__ = QuaIFLTC.__doc__ % ()
[docs]
class AdQuaIFLTC(GradNeuDyn):
r"""Adaptive quadratic integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
The adaptive quadratic integrate-and-fire neuron model [1]_ is given by:
.. math::
\begin{aligned}
\tau_m \frac{d V}{d t}&=c(V-V_{rest})(V-V_c) - w + I(t), \\
\tau_w \frac{d w}{d t}&=a(V-V_{rest}) - w,
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
**References**
.. [1] Izhikevich, E. M. (2004). Which model to use for cortical spiking
neurons?. IEEE transactions on neural networks, 15(5), 1063-1070.
.. [2] Touboul, Jonathan. "Bifurcation analysis of a general class of
nonlinear integrate-and-fire neurons." SIAM Journal on Applied
Mathematics 68, no. 4 (2008): 1045-1079.
**Examples**
Here is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdQuaIFLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== =======================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- -------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger
than :math:`V_{rest}`.
a 1 \ The sensitivity of the recovery variable :math:`u` to
the sub-threshold fluctuations of the membrane
potential :math:`v`
b .1 \ The increment of :math:`w` produced by a spike.
c .07 \ Coefficient describes membrane potential update.
Larger than 0.
tau 10 ms Membrane time constant.
tau_w 10 ms Time constant of the adaptation current.
============= ============== ======== =======================================================
**Model Variables**
================== ================= ==========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ----------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
t_last_spike -1e7 Last spike time stamp.
================== ================= ==========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -30.,
V_c: Union[float, ArrayType, Callable] = -50.0,
a: Union[float, ArrayType, Callable] = 1.,
b: Union[float, ArrayType, Callable] = .1,
c: Union[float, ArrayType, Callable] = .07,
tau: Union[float, ArrayType, Callable] = 10.,
tau_w: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
w_initializer: Union[Callable, ArrayType] = ZeroInit(),
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th = self.offset_scaling(self.init_param(V_th))
self.V_c = self.offset_scaling(self.init_param(V_c))
self.a = self.init_param(a)
self.b = self.std_scaling(self.init_param(b))
self.c = self.inv_scaling(self.init_param(c))
self.tau = self.init_param(tau)
self.tau_w = self.init_param(tau_w)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._w_initializer = is_initializer(w_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def dV(self, V, t, w, I):
I = self.sum_current_inputs(V, init=I)
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) - w + I) / self.tau
return dVdt
def dw(self, w, t, V):
dwdt = (self.a * (V - self.V_rest) - w) / self.tau_w
return dwdt
@property
def derivative(self):
return JointEq([self.dV, self.dw])
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.w = self.std_scaling(self.init_variable(self._w_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, w = self.integral(self.V.value, self.w.value, t, x, dt)
V += self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
w += self.b * spike
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
w = bm.where(spike, w + self.b, w)
self.V.value = V
self.w.value = w
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class AdQuaIF(AdQuaIFLTC):
r"""Adaptive quadratic integrate-and-fire neuron model.
**Model Descriptions**
The adaptive quadratic integrate-and-fire neuron model [1]_ is given by:
.. math::
\begin{aligned}
\tau_m \frac{d V}{d t}&=c(V-V_{rest})(V-V_c) - w + I(t), \\
\tau_w \frac{d w}{d t}&=a(V-V_{rest}) - w,
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
**References**
.. [1] Izhikevich, E. M. (2004). Which model to use for cortical spiking
neurons?. IEEE transactions on neural networks, 15(5), 1063-1070.
.. [2] Touboul, Jonathan. "Bifurcation analysis of a general class of
nonlinear integrate-and-fire neurons." SIAM Journal on Applied
Mathematics 68, no. 4 (2008): 1045-1079.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdQuaIF(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== =======================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- -------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger
than :math:`V_{rest}`.
a 1 \ The sensitivity of the recovery variable :math:`u` to
the sub-threshold fluctuations of the membrane
potential :math:`v`
b .1 \ The increment of :math:`w` produced by a spike.
c .07 \ Coefficient describes membrane potential update.
Larger than 0.
tau 10 ms Membrane time constant.
tau_w 10 ms Time constant of the adaptation current.
============= ============== ======== =======================================================
**Model Variables**
================== ================= ==========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ----------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
t_last_spike -1e7 Last spike time stamp.
================== ================= ==========================================================
Args:
%s
%s
"""
def dV(self, V, t, w, I):
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) - w + I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class AdQuaIFRefLTC(AdQuaIFLTC):
r"""Adaptive quadratic integrate-and-fire neuron model with liquid time-constant.
**Model Descriptions**
The adaptive quadratic integrate-and-fire neuron model [1]_ is given by:
.. math::
\begin{aligned}
\tau_m \frac{d V}{d t}&=c(V-V_{rest})(V-V_c) - w + I(t), \\
\tau_w \frac{d w}{d t}&=a(V-V_{rest}) - w,
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
**References**
.. [1] Izhikevich, E. M. (2004). Which model to use for cortical spiking
neurons?. IEEE transactions on neural networks, 15(5), 1063-1070.
.. [2] Touboul, Jonathan. "Bifurcation analysis of a general class of
nonlinear integrate-and-fire neurons." SIAM Journal on Applied
Mathematics 68, no. 4 (2008): 1045-1079.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdQuaIFRefLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== =======================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- -------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger
than :math:`V_{rest}`.
a 1 \ The sensitivity of the recovery variable :math:`u` to
the sub-threshold fluctuations of the membrane
potential :math:`v`
b .1 \ The increment of :math:`w` produced by a spike.
c .07 \ Coefficient describes membrane potential update.
Larger than 0.
tau 10 ms Membrane time constant.
tau_w 10 ms Time constant of the adaptation current.
============= ============== ======== =======================================================
**Model Variables**
================== ================= ==========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ----------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
t_last_spike -1e7 Last spike time stamp.
================== ================= ==========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = -65.,
V_reset: Union[float, ArrayType, Callable] = -68.,
V_th: Union[float, ArrayType, Callable] = -30.,
V_c: Union[float, ArrayType, Callable] = -50.0,
a: Union[float, ArrayType, Callable] = 1.,
b: Union[float, ArrayType, Callable] = .1,
c: Union[float, ArrayType, Callable] = .07,
tau: Union[float, ArrayType, Callable] = 10.,
tau_w: Union[float, ArrayType, Callable] = 10.,
V_initializer: Union[Callable, ArrayType] = ZeroInit(),
w_initializer: Union[Callable, ArrayType] = ZeroInit(),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th=V_th,
V_c=V_c,
a=a,
b=b,
c=c,
tau=tau,
tau_w=tau_w,
V_initializer=V_initializer,
w_initializer=w_initializer
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._w_initializer = is_initializer(w_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e8)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, w = self.integral(self.V.value, self.w.value, t, x, dt)
V += self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
w += self.b * spike_no_grad
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
w = bm.where(spike, w + self.b, w)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.w.value = w
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class AdQuaIFRef(AdQuaIFRefLTC):
r"""Adaptive quadratic integrate-and-fire neuron model.
**Model Descriptions**
The adaptive quadratic integrate-and-fire neuron model [1]_ is given by:
.. math::
\begin{aligned}
\tau_m \frac{d V}{d t}&=c(V-V_{rest})(V-V_c) - w + I(t), \\
\tau_w \frac{d w}{d t}&=a(V-V_{rest}) - w,
\end{aligned}
once the membrane potential reaches the spike threshold,
.. math::
V \rightarrow V_{reset}, \\
w \rightarrow w+b.
**References**
.. [1] Izhikevich, E. M. (2004). Which model to use for cortical spiking
neurons?. IEEE transactions on neural networks, 15(5), 1063-1070.
.. [2] Touboul, Jonathan. "Bifurcation analysis of a general class of
nonlinear integrate-and-fire neurons." SIAM Journal on Applied
Mathematics 68, no. 4 (2008): 1045-1079.
**Examples**
There is an example usage:
.. code-block:: python
import brainpy as bp
neu = bp.dyn.AdQuaIFRef(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Parameters**
============= ============== ======== =======================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- -------------------------------------------------------
V_rest -65 mV Resting potential.
V_reset -68 mV Reset potential after spike.
V_th -30 mV Threshold potential of spike and reset.
V_c -50 mV Critical voltage for spike initiation. Must be larger
than :math:`V_{rest}`.
a 1 \ The sensitivity of the recovery variable :math:`u` to
the sub-threshold fluctuations of the membrane
potential :math:`v`
b .1 \ The increment of :math:`w` produced by a spike.
c .07 \ Coefficient describes membrane potential update.
Larger than 0.
tau 10 ms Membrane time constant.
tau_w 10 ms Time constant of the adaptation current.
============= ============== ======== =======================================================
**Model Variables**
================== ================= ==========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ----------------------------------------------------------
V 0 Membrane potential.
w 0 Adaptation current.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
t_last_spike -1e7 Last spike time stamp.
================== ================= ==========================================================
Args:
%s
%s
%s
"""
def dV(self, V, t, w, I):
dVdt = (self.c * (V - self.V_rest) * (V - self.V_c) - w + I) / self.tau
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
AdQuaIF.__doc__ = AdQuaIF.__doc__ % (pneu_doc, dpneu_doc)
AdQuaIFRefLTC.__doc__ = AdQuaIFRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
AdQuaIFRef.__doc__ = AdQuaIFRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
AdQuaIFLTC.__doc__ = AdQuaIFLTC.__doc__ % ()
[docs]
class GifLTC(GradNeuDyn):
r"""Generalized Integrate-and-Fire model with liquid time-constant.
**Model Descriptions**
The generalized integrate-and-fire model [1]_ is given by
.. math::
&\frac{d I_j}{d t} = - k_j I_j
&\frac{d V}{d t} = ( - (V - V_{rest}) + R\sum_{j}I_j + RI) / \tau
&\frac{d V_{th}}{d t} = a(V - V_{rest}) - b(V_{th} - V_{th\infty})
When :math:`V` meet :math:`V_{th}`, Generalized IF neuron fires:
.. math::
&I_j \leftarrow R_j I_j + A_j
&V \leftarrow V_{reset}
&V_{th} \leftarrow max(V_{th_{reset}}, V_{th})
Note that :math:`I_j` refers to arbitrary number of internal currents.
**References**
.. [1] Mihalaş, Ştefan, and Ernst Niebur. "A generalized linear
integrate-and-fire neural model produces diverse spiking
behaviors." Neural computation 21.3 (2009): 704-718.
.. [2] Teeter, Corinne, Ramakrishnan Iyer, Vilas Menon, Nathan
Gouwens, David Feng, Jim Berg, Aaron Szafer et al. "Generalized
leaky integrate-and-fire models classify multiple neuron types."
Nature communications 9, no. 1 (2018): 1-15.
**Examples**
There is a simple usage: you r bound to be together, roy and edward
.. code-block:: python
import brainpy as bp
import matplotlib.pyplot as plt
# Tonic Spiking
neu = bp.dyn.Gif(1)
inputs = bp.inputs.ramp_input(.2, 2, 400, 0, 400)
runner = bp.DSRunner(neu, monitors=['V', 'V_th'])
runner.run(inputs=inputs)
ts = runner.mon.ts
fig, gs = bp.visualize.get_figure(1, 1, 4, 8)
ax1 = fig.add_subplot(gs[0, 0])
ax1.plot(ts, runner.mon.V[:, 0], label='V')
ax1.plot(ts, runner.mon.V_th[:, 0], label='V_th')
plt.show()
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Niebur_2009_GIF.html>`_
**Model Parameters**
============= ============== ======== ====================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------
V_rest -70 mV Resting potential.
V_reset -70 mV Reset potential after spike.
V_th_inf -50 mV Target value of threshold potential :math:`V_{th}` updating.
V_th_reset -60 mV Free parameter, should be larger than :math:`V_{reset}`.
R 20 \ Membrane resistance.
tau 20 ms Membrane time constant. Compute by :math:`R * C`.
a 0 \ Coefficient describes the dependence of
:math:`V_{th}` on membrane potential.
b 0.01 \ Coefficient describes :math:`V_{th}` update.
k1 0.2 \ Constant pf :math:`I1`.
k2 0.02 \ Constant of :math:`I2`.
R1 0 \ Free parameter.
Describes dependence of :math:`I_1` reset value on
:math:`I_1` value before spiking.
R2 1 \ Free parameter.
Describes dependence of :math:`I_2` reset value on
:math:`I_2` value before spiking.
A1 0 \ Free parameter.
A2 0 \ Free parameter.
============= ============== ======== ====================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -70 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
V_th -50 Spiking threshold potential.
I1 0 Internal current 1.
I2 0 Internal current 2.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_rest: Union[float, ArrayType, Callable] = -70.,
V_reset: Union[float, ArrayType, Callable] = -70.,
V_th_inf: Union[float, ArrayType, Callable] = -50.,
V_th_reset: Union[float, ArrayType, Callable] = -60.,
R: Union[float, ArrayType, Callable] = 20.,
tau: Union[float, ArrayType, Callable] = 20.,
a: Union[float, ArrayType, Callable] = 0.,
b: Union[float, ArrayType, Callable] = 0.01,
k1: Union[float, ArrayType, Callable] = 0.2,
k2: Union[float, ArrayType, Callable] = 0.02,
R1: Union[float, ArrayType, Callable] = 0.,
R2: Union[float, ArrayType, Callable] = 1.,
A1: Union[float, ArrayType, Callable] = 0.,
A2: Union[float, ArrayType, Callable] = 0.,
V_initializer: Union[Callable, ArrayType] = OneInit(-70.),
I1_initializer: Union[Callable, ArrayType] = ZeroInit(),
I2_initializer: Union[Callable, ArrayType] = ZeroInit(),
Vth_initializer: Union[Callable, ArrayType] = OneInit(-50.),
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_rest = self.offset_scaling(self.init_param(V_rest))
self.V_reset = self.offset_scaling(self.init_param(V_reset))
self.V_th_inf = self.offset_scaling(self.init_param(V_th_inf))
self.V_th_reset = self.offset_scaling(self.init_param(V_th_reset))
self.R = self.init_param(R)
self.a = self.init_param(a)
self.b = self.init_param(b)
self.k1 = self.init_param(k1)
self.k2 = self.init_param(k2)
self.R1 = self.init_param(R1)
self.R2 = self.init_param(R2)
self.A1 = self.std_scaling(self.init_param(A1))
self.A2 = self.std_scaling(self.init_param(A2))
self.tau = self.init_param(tau)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._I1_initializer = is_initializer(I1_initializer)
self._I2_initializer = is_initializer(I2_initializer)
self._Vth_initializer = is_initializer(Vth_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=4)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def dI1(self, I1, t):
return - self.k1 * I1
def dI2(self, I2, t):
return - self.k2 * I2
def dVth(self, V_th, t, V):
return self.a * (V - self.V_rest) - self.b * (V_th - self.V_th_inf)
def dV(self, V, t, I1, I2, I):
I = self.sum_current_inputs(V, init=I)
return (- (V - self.V_rest) + self.R * (I + I1 + I2)) / self.tau
@property
def derivative(self):
return JointEq(self.dI1, self.dI2, self.dVth, self.dV)
def reset_state(self, batch_size=None, **kwargs):
self.V = self.offset_scaling(self.init_variable(self._V_initializer, batch_size))
self.V_th = self.offset_scaling(self.init_variable(self._Vth_initializer, batch_size))
self.I1 = self.std_scaling(self.init_variable(self._I1_initializer, batch_size))
self.I2 = self.std_scaling(self.init_variable(self._I2_initializer, batch_size))
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
I1, I2, V_th, V = self.integral(self.I1.value, self.I2.value, self.V_th.value, self.V.value, t, x, dt)
V += self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike
else:
raise ValueError
I1 += spike * (self.R1 * I1 + self.A1 - I1)
I2 += spike * (self.R2 * I2 + self.A2 - I2)
V_th += (bm.maximum(self.V_th_reset, V_th) - V_th) * spike
else:
spike = self.V_th <= V
V = bm.where(spike, self.V_reset, V)
I1 = bm.where(spike, self.R1 * I1 + self.A1, I1)
I2 = bm.where(spike, self.R2 * I2 + self.A2, I2)
V_th = bm.where(spike, bm.maximum(self.V_th_reset, V_th), V_th)
self.spike.value = spike
self.I1.value = I1
self.I2.value = I2
self.V_th.value = V_th
self.V.value = V
return spike
def return_info(self):
return self.spike
[docs]
class Gif(GifLTC):
r"""Generalized Integrate-and-Fire model.
**Model Descriptions**
The generalized integrate-and-fire model [1]_ is given by
.. math::
&\frac{d I_j}{d t} = - k_j I_j
&\frac{d V}{d t} = ( - (V - V_{rest}) + R\sum_{j}I_j + RI) / \tau
&\frac{d V_{th}}{d t} = a(V - V_{rest}) - b(V_{th} - V_{th\infty})
When :math:`V` meet :math:`V_{th}`, Generalized IF neuron fires:
.. math::
&I_j \leftarrow R_j I_j + A_j
&V \leftarrow V_{reset}
&V_{th} \leftarrow max(V_{th_{reset}}, V_{th})
Note that :math:`I_j` refers to arbitrary number of internal currents.
**References**
.. [1] Mihalaş, Ştefan, and Ernst Niebur. "A generalized linear
integrate-and-fire neural model produces diverse spiking
behaviors." Neural computation 21.3 (2009): 704-718.
.. [2] Teeter, Corinne, Ramakrishnan Iyer, Vilas Menon, Nathan
Gouwens, David Feng, Jim Berg, Aaron Szafer et al. "Generalized
leaky integrate-and-fire models classify multiple neuron types."
Nature communications 9, no. 1 (2018): 1-15.
**Examples**
There is a simple usage:
.. code-block:: python
import brainpy as bp
import matplotlib.pyplot as plt
# Phasic Spiking
neu = bp.dyn.Gif(1, a=0.005)
inputs = bp.inputs.section_input((0, 1.5), (50, 500))
runner = bp.DSRunner(neu, monitors=['V', 'V_th'])
runner.run(inputs=inputs)
ts = runner.mon.ts
fig, gs = bp.visualize.get_figure(1, 1, 4, 8)
ax1 = fig.add_subplot(gs[0, 0])
ax1.plot(ts, runner.mon.V[:, 0], label='V')
ax1.plot(ts, runner.mon.V_th[:, 0], label='V_th')
plt.show()
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Niebur_2009_GIF.html>`_
**Model Parameters**
============= ============== ======== ====================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------
V_rest -70 mV Resting potential.
V_reset -70 mV Reset potential after spike.
V_th_inf -50 mV Target value of threshold potential :math:`V_{th}` updating.
V_th_reset -60 mV Free parameter, should be larger than :math:`V_{reset}`.
R 20 \ Membrane resistance.
tau 20 ms Membrane time constant. Compute by :math:`R * C`.
a 0 \ Coefficient describes the dependence of
:math:`V_{th}` on membrane potential.
b 0.01 \ Coefficient describes :math:`V_{th}` update.
k1 0.2 \ Constant pf :math:`I1`.
k2 0.02 \ Constant of :math:`I2`.
R1 0 \ Free parameter.
Describes dependence of :math:`I_1` reset value on
:math:`I_1` value before spiking.
R2 1 \ Free parameter.
Describes dependence of :math:`I_2` reset value on
:math:`I_2` value before spiking.
A1 0 \ Free parameter.
A2 0 \ Free parameter.
============= ============== ======== ====================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -70 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
V_th -50 Spiking threshold potential.
I1 0 Internal current 1.
I2 0 Internal current 2.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
"""
def dV(self, V, t, I1, I2, I):
return (- (V - self.V_rest) + self.R * (I + I1 + I2)) / self.tau
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class GifRefLTC(GifLTC):
r"""Generalized Integrate-and-Fire model with liquid time-constant.
**Model Descriptions**
The generalized integrate-and-fire model [1]_ is given by
.. math::
&\frac{d I_j}{d t} = - k_j I_j
&\frac{d V}{d t} = ( - (V - V_{rest}) + R\sum_{j}I_j + RI) / \tau
&\frac{d V_{th}}{d t} = a(V - V_{rest}) - b(V_{th} - V_{th\infty})
When :math:`V` meet :math:`V_{th}`, Generalized IF neuron fires:
.. math::
&I_j \leftarrow R_j I_j + A_j
&V \leftarrow V_{reset}
&V_{th} \leftarrow max(V_{th_{reset}}, V_{th})
Note that :math:`I_j` refers to arbitrary number of internal currents.
**References**
.. [1] Mihalaş, Ştefan, and Ernst Niebur. "A generalized linear
integrate-and-fire neural model produces diverse spiking
behaviors." Neural computation 21.3 (2009): 704-718.
.. [2] Teeter, Corinne, Ramakrishnan Iyer, Vilas Menon, Nathan
Gouwens, David Feng, Jim Berg, Aaron Szafer et al. "Generalized
leaky integrate-and-fire models classify multiple neuron types."
Nature communications 9, no. 1 (2018): 1-15.
**Examples**
There is a simple usage: mustang i love u
.. code-block:: python
import brainpy as bp
import matplotlib.pyplot as plt
# Hyperpolarization-induced Spiking
neu = bp.dyn.GifRefLTC(1, a=0.005)
neu.V_th[:] = -50.
inputs = bp.inputs.section_input((1.5, 1.7, 1.5, 1.7), (100, 400, 100, 400))
runner = bp.DSRunner(neu, monitors=['V', 'V_th'])
runner.run(inputs=inputs)
ts = runner.mon.ts
fig, gs = bp.visualize.get_figure(1, 1, 4, 8)
ax1 = fig.add_subplot(gs[0, 0])
ax1.plot(ts, runner.mon.V[:, 0], label='V')
ax1.plot(ts, runner.mon.V_th[:, 0], label='V_th')
plt.show()
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Niebur_2009_GIF.html>`_
**Model Parameters**
============= ============== ======== ====================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------
V_rest -70 mV Resting potential.
V_reset -70 mV Reset potential after spike.
V_th_inf -50 mV Target value of threshold potential :math:`V_{th}` updating.
V_th_reset -60 mV Free parameter, should be larger than :math:`V_{reset}`.
R 20 \ Membrane resistance.
tau 20 ms Membrane time constant. Compute by :math:`R * C`.
a 0 \ Coefficient describes the dependence of
:math:`V_{th}` on membrane potential.
b 0.01 \ Coefficient describes :math:`V_{th}` update.
k1 0.2 \ Constant pf :math:`I1`.
k2 0.02 \ Constant of :math:`I2`.
R1 0 \ Free parameter.
Describes dependence of :math:`I_1` reset value on
:math:`I_1` value before spiking.
R2 1 \ Free parameter.
Describes dependence of :math:`I_2` reset value on
:math:`I_2` value before spiking.
A1 0 \ Free parameter.
A2 0 \ Free parameter.
============= ============== ======== ====================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -70 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
V_th -50 Spiking threshold potential.
I1 0 Internal current 1.
I2 0 Internal current 2.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_rest: Union[float, ArrayType, Callable] = -70.,
V_reset: Union[float, ArrayType, Callable] = -70.,
V_th_inf: Union[float, ArrayType, Callable] = -50.,
V_th_reset: Union[float, ArrayType, Callable] = -60.,
R: Union[float, ArrayType, Callable] = 20.,
tau: Union[float, ArrayType, Callable] = 20.,
a: Union[float, ArrayType, Callable] = 0.,
b: Union[float, ArrayType, Callable] = 0.01,
k1: Union[float, ArrayType, Callable] = 0.2,
k2: Union[float, ArrayType, Callable] = 0.02,
R1: Union[float, ArrayType, Callable] = 0.,
R2: Union[float, ArrayType, Callable] = 1.,
A1: Union[float, ArrayType, Callable] = 0.,
A2: Union[float, ArrayType, Callable] = 0.,
V_initializer: Union[Callable, ArrayType] = OneInit(-70.),
I1_initializer: Union[Callable, ArrayType] = ZeroInit(),
I2_initializer: Union[Callable, ArrayType] = ZeroInit(),
Vth_initializer: Union[Callable, ArrayType] = OneInit(-50.),
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_rest=V_rest,
V_reset=V_reset,
V_th_inf=V_th_inf,
V_th_reset=V_th_reset,
R=R,
a=a,
b=b,
k1=k1,
k2=k2,
R1=R1,
R2=R2,
A1=A1,
A2=A2,
tau=tau,
V_initializer=V_initializer,
I1_initializer=I1_initializer,
I2_initializer=I2_initializer,
Vth_initializer=Vth_initializer,
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._I1_initializer = is_initializer(I1_initializer)
self._I2_initializer = is_initializer(I2_initializer)
self._Vth_initializer = is_initializer(Vth_initializer)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=4)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e8)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
I1, I2, V_th, V = self.integral(self.I1.value, self.I2.value, self.V_th.value, self.V.value, t, x, dt)
V += self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
if self.spk_reset == 'soft':
V -= (self.V_th - self.V_reset) * spike_no_grad
elif self.spk_reset == 'hard':
V += (self.V_reset - V) * spike_no_grad
else:
raise ValueError
I1 += spike * (self.R1 * I1 + self.A1 - I1)
I2 += spike * (self.R2 * I2 + self.A2 - I2)
V_th += (bm.maximum(self.V_th_reset, V_th) - V_th) * spike_no_grad
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.V_reset, V)
I1 = bm.where(spike, self.R1 * I1 + self.A1, I1)
I2 = bm.where(spike, self.R2 * I2 + self.A2, I2)
V_th = bm.where(spike, bm.maximum(self.V_th_reset, V_th), V_th)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.I1.value = I1
self.I2.value = I2
self.V_th.value = V_th
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class GifRef(GifRefLTC):
r"""Generalized Integrate-and-Fire model.
**Model Descriptions**
The generalized integrate-and-fire model [1]_ is given by
.. math::
&\frac{d I_j}{d t} = - k_j I_j
&\frac{d V}{d t} = ( - (V - V_{rest}) + R\sum_{j}I_j + RI) / \tau
&\frac{d V_{th}}{d t} = a(V - V_{rest}) - b(V_{th} - V_{th\infty})
When :math:`V` meet :math:`V_{th}`, Generalized IF neuron fires:
.. math::
&I_j \leftarrow R_j I_j + A_j
&V \leftarrow V_{reset}
&V_{th} \leftarrow max(V_{th_{reset}}, V_{th})
Note that :math:`I_j` refers to arbitrary number of internal currents.
**References**
.. [1] Mihalaş, Ştefan, and Ernst Niebur. "A generalized linear
integrate-and-fire neural model produces diverse spiking
behaviors." Neural computation 21.3 (2009): 704-718.
.. [2] Teeter, Corinne, Ramakrishnan Iyer, Vilas Menon, Nathan
Gouwens, David Feng, Jim Berg, Aaron Szafer et al. "Generalized
leaky integrate-and-fire models classify multiple neuron types."
Nature communications 9, no. 1 (2018): 1-15.
**Examples**
There is a simple usage:
.. code-block:: python
import brainpy as bp
import matplotlib.pyplot as plt
# Tonic Bursting
neu = bp.dyn.GifRef(1, a=0.005, A1=10., A2=-0.6)
neu.V_th[:] = -50.
inputs = bp.inputs.section_input((1.5, 1.7,), (100, 400))
runner = bp.DSRunner(neu, monitors=['V', 'V_th'])
runner.run(inputs=inputs)
ts = runner.mon.ts
fig, gs = bp.visualize.get_figure(1, 1, 4, 8)
ax1 = fig.add_subplot(gs[0, 0])
ax1.plot(ts, runner.mon.V[:, 0], label='V')
ax1.plot(ts, runner.mon.V_th[:, 0], label='V_th')
plt.show()
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Niebur_2009_GIF.html>`_
**Model Parameters**
============= ============== ======== ====================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------
V_rest -70 mV Resting potential.
V_reset -70 mV Reset potential after spike.
V_th_inf -50 mV Target value of threshold potential :math:`V_{th}` updating.
V_th_reset -60 mV Free parameter, should be larger than :math:`V_{reset}`.
R 20 \ Membrane resistance.
tau 20 ms Membrane time constant. Compute by :math:`R * C`.
a 0 \ Coefficient describes the dependence of
:math:`V_{th}` on membrane potential.
b 0.01 \ Coefficient describes :math:`V_{th}` update.
k1 0.2 \ Constant pf :math:`I1`.
k2 0.02 \ Constant of :math:`I2`.
R1 0 \ Free parameter.
Describes dependence of :math:`I_1` reset value on
:math:`I_1` value before spiking.
R2 1 \ Free parameter.
Describes dependence of :math:`I_2` reset value on
:math:`I_2` value before spiking.
A1 0 \ Free parameter.
A2 0 \ Free parameter.
============= ============== ======== ====================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -70 Membrane potential.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
V_th -50 Spiking threshold potential.
I1 0 Internal current 1.
I2 0 Internal current 2.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def dV(self, V, t, I1, I2, I):
return (- (V - self.V_rest) + self.R * (I + I1 + I2)) / self.tau
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
Gif.__doc__ = Gif.__doc__ % (pneu_doc, dpneu_doc)
GifRefLTC.__doc__ = GifRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
GifRef.__doc__ = GifRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
GifLTC.__doc__ = GifLTC.__doc__ % ()
[docs]
class IzhikevichLTC(GradNeuDyn):
r"""The Izhikevich neuron model with liquid time-constant.
**Model Descriptions**
The dynamics of the Izhikevich neuron model [1]_ [2]_ is given by:
.. math ::
\frac{d V}{d t} &= 0.04 V^{2}+5 V+140-u+I
\frac{d u}{d t} &=a(b V-u)
.. math ::
\text{if} v \geq 30 \text{mV}, \text{then}
\begin{cases} v \leftarrow c \\
u \leftarrow u+d \end{cases}
**References**
.. [1] Izhikevich, Eugene M. "Simple model of spiking neurons." IEEE
Transactions on neural networks 14.6 (2003): 1569-1572.
.. [2] Izhikevich, Eugene M. "Which model to use for cortical spiking neurons?."
IEEE transactions on neural networks 15.5 (2004): 1063-1070.
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.IzhikevichLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Izhikevich_2003_Izhikevich_model.html>`_
**Model Parameters**
============= ============== ======== ================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------------------
a 0.02 \ It determines the time scaling of
the recovery variable :math:`u`.
b 0.2 \ It describes the sensitivity of the
recovery variable :math:`u` to
the sub-threshold fluctuations of the
membrane potential :math:`v`.
c -65 \ It describes the after-spike reset value
of the membrane potential :math:`v` caused by
the fast high-threshold :math:`K^{+}`
conductance.
d 8 \ It describes after-spike reset of the
recovery variable :math:`u`
caused by slow high-threshold
:math:`Na^{+}` and :math:`K^{+}` conductance.
tau_ref 0 ms Refractory period length. [ms]
V_th 30 mV The membrane potential threshold.
============= ============== ======== ================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -65 Membrane potential.
u 1 Recovery variable.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sequence[str]] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
name: Optional[str] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# neuron parameters
V_th: Union[float, ArrayType, Callable] = 30.,
p1: Union[float, ArrayType, Callable] = 0.04,
p2: Union[float, ArrayType, Callable] = 5.,
p3: Union[float, ArrayType, Callable] = 140.,
a: Union[float, ArrayType, Callable] = 0.02,
b: Union[float, ArrayType, Callable] = 0.20,
c: Union[float, ArrayType, Callable] = -65.,
d: Union[float, ArrayType, Callable] = 8.,
tau: Union[float, ArrayType, Callable] = 10.,
R: Union[float, ArrayType, Callable] = 1.,
V_initializer: Union[Callable, ArrayType] = OneInit(-70.),
u_initializer: Union[Callable, ArrayType] = None,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(size=size,
name=name,
keep_size=keep_size,
mode=mode,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
method=method,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
scaling=scaling)
# parameters
self.V_th = self.offset_scaling(self.init_param(V_th))
self.p1 = self.inv_scaling(self.init_param(p1))
p2_scaling = self.scaling.clone(bias=-p1 * 2 * self.scaling.bias, scale=1.)
self.p2 = p2_scaling.offset_scaling(self.init_param(p2))
p3_bias = p1 * self.scaling.bias ** 2 + b * self.scaling.bias - p2 * self.scaling.bias
p3_scaling = self.scaling.clone(bias=p3_bias, scale=self.scaling.scale)
self.p3 = p3_scaling.offset_scaling(self.init_param(p3))
self.a = self.init_param(a)
self.b = self.init_param(b)
self.c = self.offset_scaling(self.init_param(c))
self.d = self.std_scaling(self.init_param(d))
self.R = self.init_param(R)
self.tau = self.init_param(tau)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._u_initializer = is_initializer(u_initializer, allow_none=True)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def dV(self, V, t, u, I):
I = self.sum_current_inputs(V, init=I)
dVdt = self.p1 * V * V + self.p2 * V + self.p3 - u + I
return dVdt
def du(self, u, t, V):
dudt = self.a * (self.b * V - u)
return dudt
@property
def derivative(self):
return JointEq([self.dV, self.du])
def reset_state(self, batch_size=None, **kwargs):
self.V = self.init_variable(self._V_initializer, batch_size)
u_initializer = OneInit(self.b * self.V) if self._u_initializer is None else self._u_initializer
self._u_initializer = is_initializer(u_initializer)
self.V = self.offset_scaling(self.V)
self.u = self.offset_scaling(self.init_variable(self._u_initializer, batch_size), bias=self.b * self.scaling.bias,
scale=self.scaling.scale)
self.spike = self.init_variable(partial(bm.zeros, dtype=self.spk_dtype), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, u = self.integral(self.V.value, self.u.value, t, x, dt)
V += self.sum_delta_inputs()
# spike, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike = stop_gradient(spike) if self.detach_spk else spike
V += spike * (self.c - V)
u += spike * self.d
else:
spike = V >= self.V_th
V = bm.where(spike, self.c, V)
u = bm.where(spike, u + self.d, u)
self.V.value = V
self.u.value = u
self.spike.value = spike
return spike
def return_info(self):
return self.spike
[docs]
class Izhikevich(IzhikevichLTC):
r"""The Izhikevich neuron model.
**Model Descriptions**
The dynamics of the Izhikevich neuron model [1]_ [2]_ is given by:
.. math ::
\frac{d V}{d t} &= 0.04 V^{2}+5 V+140-u+I
\frac{d u}{d t} &=a(b V-u)
.. math ::
\text{if} v \geq 30 \text{mV}, \text{then}
\begin{cases} v \leftarrow c \\
u \leftarrow u+d \end{cases}
**References**
.. [1] Izhikevich, Eugene M. "Simple model of spiking neurons." IEEE
Transactions on neural networks 14.6 (2003): 1569-1572.
.. [2] Izhikevich, Eugene M. "Which model to use for cortical spiking neurons?."
IEEE transactions on neural networks 15.5 (2004): 1063-1070.
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.Izhikevich(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Izhikevich_2003_Izhikevich_model.html>`_
**Model Parameters**
============= ============== ======== ================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------------------
a 0.02 \ It determines the time scaling of
the recovery variable :math:`u`.
b 0.2 \ It describes the sensitivity of the
recovery variable :math:`u` to
the sub-threshold fluctuations of the
membrane potential :math:`v`.
c -65 \ It describes the after-spike reset value
of the membrane potential :math:`v` caused by
the fast high-threshold :math:`K^{+}`
conductance.
d 8 \ It describes after-spike reset of the
recovery variable :math:`u`
caused by slow high-threshold
:math:`Na^{+}` and :math:`K^{+}` conductance.
tau_ref 0 ms Refractory period length. [ms]
V_th 30 mV The membrane potential threshold.
============= ============== ======== ================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -65 Membrane potential.
u 1 Recovery variable.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
"""
def dV(self, V, t, u, I):
dVdt = self.p1 * V * V + self.p2 * V + self.p3 - u + I
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
[docs]
class IzhikevichRefLTC(IzhikevichLTC):
r"""The Izhikevich neuron model with liquid time-constant.
**Model Descriptions**
The dynamics of the Izhikevich neuron model [1]_ [2]_ is given by:
.. math ::
\frac{d V}{d t} &= 0.04 V^{2}+5 V+140-u+I
\frac{d u}{d t} &=a(b V-u)
.. math ::
\text{if} v \geq 30 \text{mV}, \text{then}
\begin{cases} v \leftarrow c \\
u \leftarrow u+d \end{cases}
**References**
.. [1] Izhikevich, Eugene M. "Simple model of spiking neurons." IEEE
Transactions on neural networks 14.6 (2003): 1569-1572.
.. [2] Izhikevich, Eugene M. "Which model to use for cortical spiking neurons?."
IEEE transactions on neural networks 15.5 (2004): 1063-1070.
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.IzhikevichRefLTC(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Izhikevich_2003_Izhikevich_model.html>`_
**Model Parameters**
============= ============== ======== ================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------------------
a 0.02 \ It determines the time scaling of
the recovery variable :math:`u`.
b 0.2 \ It describes the sensitivity of the
recovery variable :math:`u` to
the sub-threshold fluctuations of the
membrane potential :math:`v`.
c -65 \ It describes the after-spike reset value
of the membrane potential :math:`v` caused by
the fast high-threshold :math:`K^{+}`
conductance.
d 8 \ It describes after-spike reset of the
recovery variable :math:`u`
caused by slow high-threshold
:math:`Na^{+}` and :math:`K^{+}` conductance.
tau_ref 0 ms Refractory period length. [ms]
V_th 30 mV The membrane potential threshold.
============= ============== ======== ================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -65 Membrane potential.
u 1 Recovery variable.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def __init__(
self,
size: Shape,
sharding: Optional[Sharding] = None,
keep_size: bool = False,
mode: Optional[bm.Mode] = None,
spk_fun: Callable = bm.surrogate.InvSquareGrad(),
spk_dtype: Any = None,
spk_reset: str = 'soft',
detach_spk: bool = False,
method: str = 'exp_auto',
name: Optional[str] = None,
init_var: bool = True,
scaling: Optional[bm.Scaling] = None,
# old neuron parameter
V_th: Union[float, ArrayType, Callable] = 30.,
p1: Union[float, ArrayType, Callable] = 0.04,
p2: Union[float, ArrayType, Callable] = 5.,
p3: Union[float, ArrayType, Callable] = 140.,
a: Union[float, ArrayType, Callable] = 0.02,
b: Union[float, ArrayType, Callable] = 0.20,
c: Union[float, ArrayType, Callable] = -65.,
d: Union[float, ArrayType, Callable] = 8.,
tau: Union[float, ArrayType, Callable] = 10.,
R: Union[float, ArrayType, Callable] = 1.,
V_initializer: Union[Callable, ArrayType] = OneInit(-70.),
u_initializer: Union[Callable, ArrayType] = None,
# new neuron parameter
tau_ref: Union[float, ArrayType, Callable] = 0.,
ref_var: bool = False,
# noise
noise: Union[float, ArrayType, Callable] = None,
):
# initialization
super().__init__(
size=size,
name=name,
keep_size=keep_size,
mode=mode,
method=method,
sharding=sharding,
spk_fun=spk_fun,
detach_spk=detach_spk,
spk_dtype=spk_dtype,
spk_reset=spk_reset,
init_var=False,
scaling=scaling,
V_th=V_th,
p1=p1,
p2=p2,
p3=p3,
a=a,
b=b,
c=c,
d=d,
R=R,
tau=tau,
V_initializer=V_initializer,
u_initializer=u_initializer
)
# parameters
self.ref_var = ref_var
self.tau_ref = self.init_param(tau_ref)
# initializers
self._V_initializer = is_initializer(V_initializer)
self._u_initializer = is_initializer(u_initializer, allow_none=True)
# integral
self.noise = init_noise(noise, self.varshape, num_vars=2)
if self.noise is not None:
self.integral = sdeint(method=self.method, f=self.derivative, g=self.noise)
else:
self.integral = odeint(method=method, f=self.derivative)
# variables
if init_var:
self.reset_state(self.mode)
def reset_state(self, batch_size=None, **kwargs):
super().reset_state(batch_size, **kwargs)
self.t_last_spike = self.init_variable(bm.ones, batch_size)
self.t_last_spike.fill_(-1e7)
if self.ref_var:
self.refractory = self.init_variable(partial(bm.zeros, dtype=bool), batch_size)
[docs]
def update(self, x=None):
t = share.load('t')
dt = share.load('dt')
x = 0. if x is None else x
# integrate membrane potential
V, u = self.integral(self.V.value, self.u.value, t, x, dt)
V += self.sum_delta_inputs()
# refractory
refractory = (t - self.t_last_spike) <= self.tau_ref
if isinstance(self.mode, bm.TrainingMode):
refractory = stop_gradient(refractory)
V = bm.where(refractory, self.V.value, V)
# spike, refractory, spiking time, and membrane potential reset
if isinstance(self.mode, bm.TrainingMode):
spike = self.spk_fun(V - self.V_th)
spike_no_grad = stop_gradient(spike) if self.detach_spk else spike
V += spike * (self.c - V)
u += spike * self.d
spike_ = spike_no_grad > 0.
# will be used in other place, like Delta Synapse, so stop its gradient
if self.ref_var:
self.refractory.value = stop_gradient(bm.logical_or(refractory, spike_).value)
t_last_spike = stop_gradient(bm.where(spike_, t, self.t_last_spike.value))
else:
spike = V >= self.V_th
V = bm.where(spike, self.c, V)
u = bm.where(spike, u + self.d, u)
if self.ref_var:
self.refractory.value = bm.logical_or(refractory, spike)
t_last_spike = bm.where(spike, t, self.t_last_spike.value)
self.V.value = V
self.u.value = u
self.spike.value = spike
self.t_last_spike.value = t_last_spike
return spike
[docs]
class IzhikevichRef(IzhikevichRefLTC):
r"""The Izhikevich neuron model.
**Model Descriptions**
The dynamics of the Izhikevich neuron model [1]_ [2]_ is given by:
.. math ::
\frac{d V}{d t} &= 0.04 V^{2}+5 V+140-u+I
\frac{d u}{d t} &=a(b V-u)
.. math ::
\text{if} v \geq 30 \text{mV}, \text{then}
\begin{cases} v \leftarrow c \\
u \leftarrow u+d \end{cases}
**References**
.. [1] Izhikevich, Eugene M. "Simple model of spiking neurons." IEEE
Transactions on neural networks 14.6 (2003): 1569-1572.
.. [2] Izhikevich, Eugene M. "Which model to use for cortical spiking neurons?."
IEEE transactions on neural networks 15.5 (2004): 1063-1070.
**Examples**
There is a simple usage example::
import brainpy as bp
neu = bp.dyn.IzhikevichRef(2)
# section input with wiener process
inp1 = bp.inputs.wiener_process(500., n=1, t_start=100., t_end=400.).flatten()
inputs = bp.inputs.section_input([0., 22., 0.], [100., 300., 100.]) + inp1
runner = bp.DSRunner(neu, monitors=['V'])
runner.run(inputs=inputs)
bp.visualize.line_plot(runner.mon['ts'], runner.mon['V'], plot_ids=(0, 1), show=True)
**Model Examples**
- `Detailed examples to reproduce different firing patterns <https://brainpy-examples.readthedocs.io/en/latest/neurons/Izhikevich_2003_Izhikevich_model.html>`_
**Model Parameters**
============= ============== ======== ================================================================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- --------------------------------------------------------------------------------
a 0.02 \ It determines the time scaling of
the recovery variable :math:`u`.
b 0.2 \ It describes the sensitivity of the
recovery variable :math:`u` to
the sub-threshold fluctuations of the
membrane potential :math:`v`.
c -65 \ It describes the after-spike reset value
of the membrane potential :math:`v` caused by
the fast high-threshold :math:`K^{+}`
conductance.
d 8 \ It describes after-spike reset of the
recovery variable :math:`u`
caused by slow high-threshold
:math:`Na^{+}` and :math:`K^{+}` conductance.
tau_ref 0 ms Refractory period length. [ms]
V_th 30 mV The membrane potential threshold.
============= ============== ======== ================================================================================
**Model Variables**
================== ================= =========================================================
**Variables name** **Initial Value** **Explanation**
------------------ ----------------- ---------------------------------------------------------
V -65 Membrane potential.
u 1 Recovery variable.
input 0 External and synaptic input current.
spike False Flag to mark whether the neuron is spiking.
refractory False Flag to mark whether the neuron is in refractory period.
t_last_spike -1e7 Last spike time stamp.
================== ================= =========================================================
Args:
%s
%s
%s
"""
def dV(self, V, t, u, I):
dVdt = self.p1 * V * V + self.p2 * V + self.p3 - u + I
return dVdt
[docs]
def update(self, x=None):
x = 0. if x is None else x
x = self.sum_current_inputs(self.V.value, init=x)
return super().update(x)
Izhikevich.__doc__ = Izhikevich.__doc__ % (pneu_doc, dpneu_doc)
IzhikevichRefLTC.__doc__ = IzhikevichRefLTC.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
IzhikevichRef.__doc__ = IzhikevichRef.__doc__ % (pneu_doc, dpneu_doc, ref_doc)
IzhikevichLTC.__doc__ = IzhikevichLTC.__doc__ % ()
![[docs]](../../../../../apis/generated/brainpy.dyn.Gif.html#brainpy.dyn.Gif){kind=link}
![[docs]](../../../../../apis/generated/brainpy.dyn.Gif.html#brainpy.dyn.Gif.update){kind=link}