# -*- coding: utf-8 -*-
from typing import Union, Dict, Callable, Optional
from jax import vmap
from jax.lax import stop_gradient
import brainpy.math as bm
from brainpy.connect import TwoEndConnector, All2All, One2One
from brainpy.dyn.base import NeuGroup, SynOut, SynSTP, TwoEndConn
from brainpy.initialize import Initializer, variable
from brainpy.integrators import odeint, JointEq
from brainpy.modes import Mode, BatchingMode, normal
from brainpy.types import Array
from ..synouts import CUBA, MgBlock
__all__ = [
'Delta',
'Exponential',
'DualExponential',
'Alpha',
'NMDA',
]
[docs]class Delta(TwoEndConn):
r"""Voltage Jump Synapse Model, or alias of Delta Synapse Model.
**Model Descriptions**
.. math::
I_{syn} (t) = \sum_{j\in C} g_{\mathrm{max}} * \mathrm{STP} * \delta(t-t_j-D)
where :math:`g_{\mathrm{max}}` denotes the chemical synaptic strength,
:math:`t_j` the spiking moment of the presynaptic neuron :math:`j`,
:math:`C` the set of neurons connected to the post-synaptic neuron,
:math:`D` the transmission delay of chemical synapses,
and :math:`\mathrm{STP}` the short-term plasticity effect.
For simplicity, the rise and decay phases of post-synaptic currents are
omitted in this model.
**Model Examples**
.. plot::
:include-source: True
>>> import brainpy as bp
>>> from brainpy.dyn import synapses, neurons
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = neurons.LIF(1)
>>> neu2 = neurons.LIF(1)
>>> syn1 = synapses.Alpha(neu1, neu2, bp.connect.All2All(), weights=5.)
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 25.), ('post.input', 10.)], monitors=['pre.V', 'post.V', 'pre.spike'])
>>> runner.run(150.)
>>>
>>> fig, gs = bp.visualize.get_figure(1, 1, 3, 8)
>>> plt.plot(runner.mon.ts, runner.mon['pre.V'], label='pre-V')
>>> plt.plot(runner.mon.ts, runner.mon['post.V'], label='post-V')
>>> plt.xlim(40, 150)
>>> plt.legend()
>>> plt.show()
Parameters
----------
pre: NeuGroup
The pre-synaptic neuron group.
post: NeuGroup
The post-synaptic neuron group.
conn: optional, ndarray, JaxArray, dict of (str, ndarray), TwoEndConnector
The synaptic connections.
comp_method: str
The connection type used for model speed optimization. It can be
`sparse` and `dense`. The default is `sparse`.
delay_step: int, ndarray, JaxArray, Initializer, Callable
The delay length. It should be the value of :math:`\mathrm{delay\_time / dt}`.
g_max: float, ndarray, JaxArray, Initializer, Callable
The synaptic strength. Default is 1.
post_ref_key: str
Whether the post-synaptic group has refractory period.
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Array, Dict[str, Array]],
output: SynOut = CUBA(target_var='V'),
stp: Optional[SynSTP] = None,
comp_method: str = 'sparse',
g_max: Union[float, Array, Initializer, Callable] = 1.,
delay_step: Union[float, Array, Initializer, Callable] = None,
post_ref_key: str = None,
# other parameters
name: str = None,
mode: Mode = normal,
stop_spike_gradient: bool = False,
):
super(Delta, self).__init__(name=name,
pre=pre,
post=post,
conn=conn,
output=output,
stp=stp,
mode=mode)
# parameters
self.stop_spike_gradient = stop_spike_gradient
self.post_ref_key = post_ref_key
if post_ref_key:
self.check_post_attrs(post_ref_key)
self.comp_method = comp_method
# connections and weights
self.g_max, self.conn_mask = self.init_weights(g_max, comp_method=comp_method, sparse_data='csr')
# register delay
self.delay_step = self.register_delay(f"{self.pre.name}.spike", delay_step, self.pre.spike)
def reset_state(self, batch_size=None):
self.output.reset_state(batch_size)
if self.stp is not None: self.stp.reset_state(batch_size)
def update(self, tdi, pre_spike=None):
# pre-synaptic spikes
if pre_spike is None:
pre_spike = self.get_delay_data(f"{self.pre.name}.spike", delay_step=self.delay_step)
if self.stop_spike_gradient:
pre_spike = pre_spike.value if isinstance(pre_spike, bm.JaxArray) else pre_spike
pre_spike = stop_gradient(pre_spike)
# update sub-components
self.output.update(tdi)
if self.stp is not None: self.stp.update(tdi, pre_spike)
# synaptic values onto the post
if isinstance(self.conn, All2All):
syn_value = self.stp(bm.asarray(pre_spike, dtype=bm.dftype()))
post_vs = self.syn2post_with_all2all(syn_value, self.g_max)
elif isinstance(self.conn, One2One):
syn_value = self.stp(bm.asarray(pre_spike, dtype=bm.dftype()))
post_vs = self.syn2post_with_one2one(syn_value, self.g_max)
else:
if self.comp_method == 'sparse':
f = lambda s: bm.pre2post_event_sum(s, self.conn_mask, self.post.num, self.g_max)
if isinstance(self.mode, BatchingMode): f = vmap(f)
post_vs = f(pre_spike)
# if not isinstance(self.stp, _NullSynSTP):
# raise NotImplementedError()
# stp_value = self.stp(1.)
# f2 = lambda s: bm.pre2post_sum(s, self.post.num, *self.conn_mask)
# if self.trainable: f2 = vmap(f2)
# post_vs *= f2(stp_value)
else:
syn_value = self.stp(bm.asarray(pre_spike, dtype=bm.dftype()))
post_vs = self.syn2post_with_dense(syn_value, self.g_max, self.conn_mask)
if self.post_ref_key:
post_vs = post_vs * (1. - getattr(self.post, self.post_ref_key))
# update outputs
return self.output(post_vs)
[docs]class Exponential(TwoEndConn):
r"""Exponential decay synapse model.
**Model Descriptions**
The single exponential decay synapse model assumes the release of neurotransmitter,
its diffusion across the cleft, the receptor binding, and channel opening all happen
very quickly, so that the channels instantaneously jump from the closed to the open state.
Therefore, its expression is given by
.. math::
g_{\mathrm{syn}}(t)=g_{\mathrm{max}} e^{-\left(t-t_{0}\right) / \tau}
where :math:`\tau_{delay}` is the time constant of the synaptic state decay,
:math:`t_0` is the time of the pre-synaptic spike,
:math:`g_{\mathrm{max}}` is the maximal conductance.
Accordingly, the differential form of the exponential synapse is given by
.. math::
\begin{aligned}
& g_{\mathrm{syn}}(t) = g_{max} g * \mathrm{STP} \\
& \frac{d g}{d t} = -\frac{g}{\tau_{decay}}+\sum_{k} \delta(t-t_{j}^{k}).
\end{aligned}
where :math:`\mathrm{STP}` is used to model the short-term plasticity effect.
**Model Examples**
- `(Brunel & Hakim, 1999) Fast Global Oscillation <https://brainpy-examples.readthedocs.io/en/latest/oscillation_synchronization/Brunel_Hakim_1999_fast_oscillation.html>`_
- `(Vreeswijk & Sompolinsky, 1996) E/I balanced network <https://brainpy-examples.readthedocs.io/en/latest/ei_nets/Vreeswijk_1996_EI_net.html>`_
- `(Brette, et, al., 2007) CUBA <https://brainpy-examples.readthedocs.io/en/latest/ei_nets/Brette_2007_CUBA.html>`_
- `(Tian, et al., 2020) E/I Net for fast response <https://brainpy-examples.readthedocs.io/en/latest/ei_nets/Tian_2020_EI_net_for_fast_response.html>`_
.. plot::
:include-source: True
>>> import brainpy as bp
>>> from brainpy.dyn import neurons, synapses, synouts
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = neurons.LIF(1)
>>> neu2 = neurons.LIF(1)
>>> syn1 = synapses.Exponential(neu1, neu2, bp.conn.All2All(),
>>> g_max=5., output=synouts.CUBA())
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 25.)], monitors=['pre.V', 'post.V', 'syn.g'])
>>> runner.run(150.)
>>>
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 8)
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['pre.V'], label='pre-V')
>>> plt.plot(runner.mon.ts, runner.mon['post.V'], label='post-V')
>>> plt.legend()
>>>
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.g'], label='g')
>>> plt.legend()
>>> plt.show()
Parameters
----------
pre: NeuGroup
The pre-synaptic neuron group.
post: NeuGroup
The post-synaptic neuron group.
conn: optional, ndarray, JaxArray, dict of (str, ndarray), TwoEndConnector
The synaptic connections.
comp_method: str
The connection type used for model speed optimization. It can be
`sparse` and `dense`. The default is `sparse`.
delay_step: int, ndarray, JaxArray, Initializer, Callable
The delay length. It should be the value of :math:`\mathrm{delay\_time / dt}`.
tau: float, JaxArray, ndarray
The time constant of decay. [ms]
g_max: float, ndarray, JaxArray, Initializer, Callable
The synaptic strength (the maximum conductance). Default is 1.
name: str
The name of this synaptic projection.
method: str
The numerical integration methods.
References
----------
.. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
"The Synapse." Principles of Computational Modelling in Neuroscience.
Cambridge: Cambridge UP, 2011. 172-95. Print.
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Array, Dict[str, Array]],
output: SynOut = CUBA(),
stp: Optional[SynSTP] = None,
comp_method: str = 'sparse',
g_max: Union[float, Array, Initializer, Callable] = 1.,
delay_step: Union[int, Array, Initializer, Callable] = None,
tau: Union[float, Array] = 8.0,
method: str = 'exp_auto',
# other parameters
name: str = None,
mode: Mode = normal,
stop_spike_gradient: bool = False,
):
super(Exponential, self).__init__(pre=pre,
post=post,
conn=conn,
output=output,
stp=stp,
name=name,
mode=mode)
# parameters
self.stop_spike_gradient = stop_spike_gradient
self.comp_method = comp_method
self.tau = tau
if bm.size(self.tau) != 1:
raise ValueError(f'"tau" must be a scalar or a tensor with size of 1. But we got {self.tau}')
# connections and weights
self.g_max, self.conn_mask = self.init_weights(g_max, comp_method, sparse_data='csr')
# variables
self.g = variable(bm.zeros, mode, self.post.num)
self.delay_step = self.register_delay(f"{self.pre.name}.spike", delay_step, self.pre.spike)
# function
self.integral = odeint(lambda g, t: -g / self.tau, method=method)
def reset_state(self, batch_size=None):
self.g.value = variable(bm.zeros, batch_size, self.post.num)
self.output.reset_state(batch_size)
if self.stp is not None: self.stp.reset_state(batch_size)
def update(self, tdi, pre_spike=None):
t, dt = tdi['t'], tdi['dt']
# delays
if pre_spike is None:
pre_spike = self.get_delay_data(f"{self.pre.name}.spike", self.delay_step)
if self.stop_spike_gradient:
pre_spike = pre_spike.value if isinstance(pre_spike, bm.JaxArray) else pre_spike
pre_spike = stop_gradient(pre_spike)
# update sub-components
self.output.update(tdi)
if self.stp is not None: self.stp.update(tdi, pre_spike)
# post values
if isinstance(self.conn, All2All):
syn_value = bm.asarray(pre_spike, dtype=bm.dftype())
if self.stp is not None: syn_value = self.stp(syn_value)
post_vs = self.syn2post_with_all2all(syn_value, self.g_max)
elif isinstance(self.conn, One2One):
syn_value = bm.asarray(pre_spike, dtype=bm.dftype())
if self.stp is not None: syn_value = self.stp(syn_value)
post_vs = self.syn2post_with_one2one(syn_value, self.g_max)
else:
if self.comp_method == 'sparse':
f = lambda s: bm.pre2post_event_sum(s, self.conn_mask, self.post.num, self.g_max)
if isinstance(self.mode, BatchingMode): f = vmap(f)
post_vs = f(pre_spike)
# if not isinstance(self.stp, _NullSynSTP):
# raise NotImplementedError()
else:
syn_value = bm.asarray(pre_spike, dtype=bm.dftype())
if self.stp is not None: syn_value = self.stp(syn_value)
post_vs = self.syn2post_with_dense(syn_value, self.g_max, self.conn_mask)
# updates
self.g.value = self.integral(self.g.value, t, dt) + post_vs
# output
return self.output(self.g)
[docs]class DualExponential(TwoEndConn):
r"""Dual exponential synapse model.
**Model Descriptions**
The dual exponential synapse model [1]_, also named as *difference of two exponentials* model,
is given by:
.. math::
g_{\mathrm{syn}}(t)=g_{\mathrm{max}} \frac{\tau_{1} \tau_{2}}{
\tau_{1}-\tau_{2}}\left(\exp \left(-\frac{t-t_{0}}{\tau_{1}}\right)
-\exp \left(-\frac{t-t_{0}}{\tau_{2}}\right)\right)
where :math:`\tau_1` is the time constant of the decay phase, :math:`\tau_2`
is the time constant of the rise phase, :math:`t_0` is the time of the pre-synaptic
spike, :math:`g_{\mathrm{max}}` is the maximal conductance.
However, in practice, this formula is hard to implement. The equivalent solution is
two coupled linear differential equations [2]_:
.. math::
\begin{aligned}
&g_{\mathrm{syn}}(t)=g_{\mathrm{max}} g * \mathrm{STP} \\
&\frac{d g}{d t}=-\frac{g}{\tau_{\mathrm{decay}}}+h \\
&\frac{d h}{d t}=-\frac{h}{\tau_{\text {rise }}}+ \delta\left(t_{0}-t\right),
\end{aligned}
where :math:`\mathrm{STP}` is used to model the short-term plasticity effect of synapses.
**Model Examples**
.. plot::
:include-source: True
>>> import brainpy as bp
>>> from brainpy.dyn import neurons, synapses, synouts
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = neurons.LIF(1)
>>> neu2 = neurons.LIF(1)
>>> syn1 = synapses.DualExponential(neu1, neu2, bp.connect.All2All(), output=synouts.CUBA())
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 25.)], monitors=['pre.V', 'post.V', 'syn.g', 'syn.h'])
>>> runner.run(150.)
>>>
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 8)
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['pre.V'], label='pre-V')
>>> plt.plot(runner.mon.ts, runner.mon['post.V'], label='post-V')
>>> plt.legend()
>>>
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.g'], label='g')
>>> plt.plot(runner.mon.ts, runner.mon['syn.h'], label='h')
>>> plt.legend()
>>> plt.show()
Parameters
----------
pre: NeuGroup
The pre-synaptic neuron group.
post: NeuGroup
The post-synaptic neuron group.
conn: optional, ndarray, JaxArray, dict of (str, ndarray), TwoEndConnector
The synaptic connections.
comp_method: str
The connection type used for model speed optimization. It can be
`sparse` and `dense`. The default is `sparse`.
delay_step: int, ndarray, JaxArray, Initializer, Callable
The delay length. It should be the value of :math:`\mathrm{delay\_time / dt}`.
tau_decay: float, JaxArray, JaxArray, ndarray
The time constant of the synaptic decay phase. [ms]
tau_rise: float, JaxArray, JaxArray, ndarray
The time constant of the synaptic rise phase. [ms]
g_max: float, ndarray, JaxArray, Initializer, Callable
The synaptic strength (the maximum conductance). Default is 1.
name: str
The name of this synaptic projection.
method: str
The numerical integration methods.
References
----------
.. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
"The Synapse." Principles of Computational Modelling in Neuroscience.
Cambridge: Cambridge UP, 2011. 172-95. Print.
.. [2] Roth, A., & Van Rossum, M. C. W. (2009). Modeling Synapses. Computational
Modeling Methods for Neuroscientists.
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Array, Dict[str, Array]],
stp: Optional[SynSTP] = None,
output: SynOut = CUBA(),
comp_method: str = 'dense',
g_max: Union[float, Array, Initializer, Callable] = 1.,
tau_decay: Union[float, Array] = 10.0,
tau_rise: Union[float, Array] = 1.,
delay_step: Union[int, Array, Initializer, Callable] = None,
method: str = 'exp_auto',
# other parameters
name: str = None,
mode: Mode = normal,
stop_spike_gradient: bool = False,
):
super(DualExponential, self).__init__(pre=pre,
post=post,
conn=conn,
output=output,
stp=stp,
name=name,
mode=mode)
# parameters
# self.check_pre_attrs('spike')
self.check_post_attrs('input')
self.stop_spike_gradient = stop_spike_gradient
self.comp_method = comp_method
self.tau_rise = tau_rise
self.tau_decay = tau_decay
if bm.size(self.tau_rise) != 1:
raise ValueError(f'"tau_rise" must be a scalar or a tensor with size of 1. '
f'But we got {self.tau_rise}')
if bm.size(self.tau_decay) != 1:
raise ValueError(f'"tau_decay" must be a scalar or a tensor with size of 1. '
f'But we got {self.tau_decay}')
# connections
self.g_max, self.conn_mask = self.init_weights(g_max, comp_method, sparse_data='ij')
# variables
self.h = variable(bm.zeros, mode, self.pre.num)
self.g = variable(bm.zeros, mode, self.pre.num)
self.delay_step = self.register_delay(f"{self.pre.name}.spike", delay_step, self.pre.spike)
# integral
self.integral = odeint(method=method, f=JointEq([self.dg, self.dh]))
def reset_state(self, batch_size=None):
self.h.value = variable(bm.zeros, batch_size, self.pre.num)
self.g.value = variable(bm.zeros, batch_size, self.pre.num)
self.output.reset_state(batch_size)
if self.stp is not None: self.stp.reset_state(batch_size)
def dh(self, h, t):
return -h / self.tau_rise
def dg(self, g, t, h):
return -g / self.tau_decay + h
def update(self, tdi, pre_spike=None):
t, dt = tdi['t'], tdi['dt']
# pre-synaptic spikes
if pre_spike is None:
pre_spike = self.get_delay_data(f"{self.pre.name}.spike", self.delay_step)
if self.stop_spike_gradient:
pre_spike = pre_spike.value if isinstance(pre_spike, bm.JaxArray) else pre_spike
pre_spike = stop_gradient(pre_spike)
# update sub-components
self.output.update(tdi)
if self.stp is not None: self.stp.update(tdi, pre_spike)
# update synaptic variables
self.g.value, self.h.value = self.integral(self.g, self.h, t, dt)
self.h += pre_spike
# post values
syn_value = self.g.value
if self.stp is not None: syn_value = self.stp(syn_value)
if isinstance(self.conn, All2All):
post_vs = self.syn2post_with_all2all(syn_value, self.g_max)
elif isinstance(self.conn, One2One):
post_vs = self.syn2post_with_one2one(syn_value, self.g_max)
else:
if self.comp_method == 'sparse':
f = lambda s: bm.pre2post_sum(s, self.post.num, *self.conn_mask)
if isinstance(self.mode, BatchingMode): f = vmap(f)
post_vs = f(syn_value)
else:
post_vs = self.syn2post_with_dense(syn_value, self.g_max, self.conn_mask)
# output
return self.output(post_vs)
[docs]class Alpha(DualExponential):
r"""Alpha synapse model.
**Model Descriptions**
The analytical expression of alpha synapse is given by:
.. math::
g_{syn}(t)= g_{max} \frac{t-t_{s}}{\tau} \exp \left(-\frac{t-t_{s}}{\tau}\right).
While, this equation is hard to implement. So, let's try to convert it into the
differential forms:
.. math::
\begin{aligned}
&g_{\mathrm{syn}}(t)= g_{\mathrm{max}} g \\
&\frac{d g}{d t}=-\frac{g}{\tau}+h \\
&\frac{d h}{d t}=-\frac{h}{\tau}+\delta\left(t_{0}-t\right)
\end{aligned}
**Model Examples**
.. plot::
:include-source: True
>>> import brainpy as bp
>>> from brainpy.dyn import neurons, synapses, synouts
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = neurons.LIF(1)
>>> neu2 = neurons.LIF(1)
>>> syn1 = synapses.Alpha(neu1, neu2, bp.connect.All2All(), output=synouts.CUBA())
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 25.)], monitors=['pre.V', 'post.V', 'syn.g', 'syn.h'])
>>> runner.run(150.)
>>>
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 8)
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['pre.V'], label='pre-V')
>>> plt.plot(runner.mon.ts, runner.mon['post.V'], label='post-V')
>>> plt.legend()
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.g'], label='g')
>>> plt.plot(runner.mon.ts, runner.mon['syn.h'], label='h')
>>> plt.legend()
>>> plt.show()
Parameters
----------
pre: NeuGroup
The pre-synaptic neuron group.
post: NeuGroup
The post-synaptic neuron group.
conn: optional, ndarray, JaxArray, dict of (str, ndarray), TwoEndConnector
The synaptic connections.
comp_method: str
The connection type used for model speed optimization. It can be
`sparse` and `dense`. The default is `sparse`.
delay_step: int, ndarray, JaxArray, Initializer, Callable
The delay length. It should be the value of :math:`\mathrm{delay\_time / dt}`.
tau_decay: float, JaxArray, ndarray
The time constant of the synaptic decay phase. [ms]
g_max: float, ndarray, JaxArray, Initializer, Callable
The synaptic strength (the maximum conductance). Default is 1.
name: str
The name of this synaptic projection.
method: str
The numerical integration methods.
References
----------
.. [1] Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw.
"The Synapse." Principles of Computational Modelling in Neuroscience.
Cambridge: Cambridge UP, 2011. 172-95. Print.
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Array, Dict[str, Array]],
output: SynOut = CUBA(),
stp: Optional[SynSTP] = None,
comp_method: str = 'dense',
g_max: Union[float, Array, Initializer, Callable] = 1.,
delay_step: Union[int, Array, Initializer, Callable] = None,
tau_decay: Union[float, Array] = 10.0,
method: str = 'exp_auto',
# other parameters
name: str = None,
mode: Mode = normal,
stop_spike_gradient: bool = False,
):
super(Alpha, self).__init__(pre=pre,
post=post,
conn=conn,
comp_method=comp_method,
delay_step=delay_step,
g_max=g_max,
tau_decay=tau_decay,
tau_rise=tau_decay,
method=method,
output=output,
stp=stp,
name=name,
mode=mode,
stop_spike_gradient=stop_spike_gradient)
[docs]class NMDA(TwoEndConn):
r"""NMDA synapse model.
**Model Descriptions**
The NMDA receptor is a glutamate receptor and ion channel found in neurons.
The NMDA receptor is one of three types of ionotropic glutamate receptors,
the other two being AMPA and kainate receptors.
The NMDA receptor mediated conductance depends on the postsynaptic voltage.
The voltage dependence is due to the blocking of the pore of the NMDA receptor
from the outside by a positively charged magnesium ion. The channel is
nearly completely blocked at resting potential, but the magnesium block is
relieved if the cell is depolarized. The fraction of channels :math:`g_{\infty}`
that are not blocked by magnesium can be fitted to
.. math::
g_{\infty}(V,[{Mg}^{2+}]_{o}) = (1+{e}^{-\alpha V}
\frac{[{Mg}^{2+}]_{o}} {\beta})^{-1}
Here :math:`[{Mg}^{2+}]_{o}` is the extracellular magnesium concentration,
usually 1 mM. Thus, the channel acts as a
"coincidence detector" and only once both of these conditions are met, the
channel opens and it allows positively charged ions (cations) to flow through
the cell membrane [2]_.
If we make the approximation that the magnesium block changes
instantaneously with voltage and is independent of the gating of the channel,
the net NMDA receptor-mediated synaptic current is given by
.. math::
I_{syn} = g_\mathrm{NMDA}(t) (V(t)-E) \cdot g_{\infty}
where :math:`V(t)` is the post-synaptic neuron potential, :math:`E` is the
reversal potential.
Simultaneously, the kinetics of synaptic state :math:`g` is given by
.. math::
& g_\mathrm{NMDA} (t) = g_{max} g \\
& \frac{d g}{dt} = -\frac{g} {\tau_{decay}}+a x(1-g) \\
& \frac{d x}{dt} = -\frac{x}{\tau_{rise}}+ \sum_{k} \delta(t-t_{j}^{k})
where the decay time of NMDA currents is usually taken to be
:math:`\tau_{decay}` =100 ms, :math:`a= 0.5 ms^{-1}`, and :math:`\tau_{rise}` =2 ms.
The NMDA receptor has been thought to be very important for controlling
synaptic plasticity and mediating learning and memory functions [3]_.
**Model Examples**
- `(Wang, 2002) Decision making spiking model <https://brainpy-examples.readthedocs.io/en/latest/decision_making/Wang_2002_decision_making_spiking.html>`_
.. plot::
:include-source: True
>>> import brainpy as bp
>>> from brainpy.dyn import synapses, neurons
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = neurons.HH(1)
>>> neu2 = neurons.HH(1)
>>> syn1 = synapses.NMDA(neu1, neu2, bp.connect.All2All(), E=0.)
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 5.)], monitors=['pre.V', 'post.V', 'syn.g', 'syn.x'])
>>> runner.run(150.)
>>>
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 8)
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['pre.V'], label='pre-V')
>>> plt.plot(runner.mon.ts, runner.mon['post.V'], label='post-V')
>>> plt.legend()
>>>
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.g'], label='g')
>>> plt.plot(runner.mon.ts, runner.mon['syn.x'], label='x')
>>> plt.legend()
>>> plt.show()
Parameters
----------
pre: NeuGroup
The pre-synaptic neuron group.
post: NeuGroup
The post-synaptic neuron group.
conn: optional, ndarray, JaxArray, dict of (str, ndarray), TwoEndConnector
The synaptic connections.
comp_method: str
The connection type used for model speed optimization. It can be
`sparse` and `dense`. The default is `dense`.
delay_step: int, ndarray, JaxArray, Initializer, Callable
The delay length. It should be the value of :math:`\mathrm{delay\_time / dt}`.
g_max: float, ndarray, JaxArray, Initializer, Callable
The synaptic strength (the maximum conductance). Default is 1.
tau_decay: float, JaxArray, ndarray
The time constant of the synaptic decay phase. Default 100 [ms]
tau_rise: float, JaxArray, ndarray
The time constant of the synaptic rise phase. Default 2 [ms]
a: float, JaxArray, ndarray
Default 0.5 ms^-1.
name: str
The name of this synaptic projection.
method: str
The numerical integration methods.
References
----------
.. [1] Brunel N, Wang X J. Effects of neuromodulation in a
cortical network model of object working memory dominated
by recurrent inhibition[J].
Journal of computational neuroscience, 2001, 11(1): 63-85.
.. [2] Furukawa, Hiroyasu, Satinder K. Singh, Romina Mancusso, and
Eric Gouaux. "Subunit arrangement and function in NMDA receptors."
Nature 438, no. 7065 (2005): 185-192.
.. [3] Li, F. and Tsien, J.Z., 2009. Memory and the NMDA receptors. The New
England journal of medicine, 361(3), p.302.
.. [4] https://en.wikipedia.org/wiki/NMDA_receptor
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Array, Dict[str, Array]],
output: SynOut = MgBlock(E=0., alpha=0.062, beta=3.57, cc_Mg=1.2),
stp: Optional[SynSTP] = None,
comp_method: str = 'dense',
g_max: Union[float, Array, Initializer, Callable] = 0.15,
delay_step: Union[int, Array, Initializer, Callable] = None,
tau_decay: Union[float, Array] = 100.,
a: Union[float, Array] = 0.5,
tau_rise: Union[float, Array] = 2.,
method: str = 'exp_auto',
# other parameters
name: str = None,
mode: Mode = normal,
stop_spike_gradient: bool = False,
):
super(NMDA, self).__init__(pre=pre,
post=post,
conn=conn,
output=output,
stp=stp,
name=name,
mode=mode)
# parameters
# self.check_post_attrs('input', 'V')
self.tau_decay = tau_decay
self.tau_rise = tau_rise
self.a = a
if bm.size(a) != 1:
raise ValueError(f'"a" must be a scalar or a tensor with size of 1. But we got {a}')
if bm.size(tau_decay) != 1:
raise ValueError(f'"tau_decay" must be a scalar or a tensor with size of 1. But we got {tau_decay}')
if bm.size(tau_rise) != 1:
raise ValueError(f'"tau_rise" must be a scalar or a tensor with size of 1. But we got {tau_rise}')
self.comp_method = comp_method
self.stop_spike_gradient = stop_spike_gradient
# connections and weights
self.g_max, self.conn_mask = self.init_weights(g_max, comp_method, sparse_data='ij')
# variables
self.g = variable(bm.zeros, mode, self.pre.num)
self.x = variable(bm.zeros, mode, self.pre.num)
self.delay_step = self.register_delay(f"{self.pre.name}.spike", delay_step, self.pre.spike)
# integral
self.integral = odeint(method=method, f=JointEq(self.dg, self.dx))
def dg(self, g, t, x):
return -g / self.tau_decay + self.a * x * (1 - g)
def dx(self, x, t):
return -x / self.tau_rise
def reset_state(self, batch_size=None):
self.g.value = variable(bm.zeros, batch_size, self.pre.num)
self.x.value = variable(bm.zeros, batch_size, self.pre.num)
self.output.reset_state(batch_size)
if self.stp is not None: self.stp.reset_state(batch_size)
def update(self, tdi, pre_spike=None):
t, dt = tdi['t'], tdi['dt']
# delays
if pre_spike is None:
pre_spike = self.get_delay_data(f"{self.pre.name}.spike", self.delay_step)
if self.stop_spike_gradient:
pre_spike = pre_spike.value if isinstance(pre_spike, bm.JaxArray) else pre_spike
pre_spike = stop_gradient(pre_spike)
# update sub-components
self.output.update(tdi)
if self.stp is not None: self.stp.update(tdi, pre_spike)
# update synapse variables
self.g.value, self.x.value = self.integral(self.g, self.x, t, dt=dt)
self.x += pre_spike
# post-synaptic value
syn_value = self.g.value
if self.stp is not None: syn_value = self.stp(syn_value)
if isinstance(self.conn, All2All):
post_vs = self.syn2post_with_all2all(syn_value, self.g_max)
elif isinstance(self.conn, One2One):
post_vs = self.syn2post_with_one2one(syn_value, self.g_max)
else:
if self.comp_method == 'sparse':
f = lambda s: bm.pre2post_sum(s, self.post.num, *self.conn_mask)
if isinstance(self.mode, BatchingMode): f = vmap(f)
post_vs = f(syn_value)
else:
post_vs = self.syn2post_with_dense(syn_value, self.g_max, self.conn_mask)
# output
return self.output(post_vs)
