# -*- coding: utf-8 -*-
from typing import Union, Dict, Callable
import brainpy.math as bm
from brainpy.connect import TwoEndConnector
from brainpy.dyn.base import NeuGroup, TwoEndConn
from brainpy.initialize import Initializer
from brainpy.dyn.utils import init_delay
from brainpy.integrators import odeint, JointEq
from brainpy.types import Tensor, Parameter
__all__ = [
'STP'
]
[docs]class STP(TwoEndConn):
r"""Short-term plasticity model.
**Model Descriptions**
Short-term plasticity (STP) [1]_ [2]_ [3]_, also called dynamical synapses,
refers to the changes of synaptic strengths over time in a way that reflects
the history of presynaptic activity. Two types of STP, with opposite effects
on synaptic efficacy, have been observed in experiments. They are known as
Short-Term Depression (STD) and Short-Term Facilitation (STF).
In the model proposed by Tsodyks and Markram [4]_ [5]_, the STD effect is
modeled by a normalized variable :math:`x (0 \le x \le 1)`, denoting the fraction
of resources that remain available after neurotransmitter depletion.
The STF effect is modeled by a utilization parameter :math:`u`, representing
the fraction of available resources ready for use (release probability).
Following a spike,
- (i) :math:`u` increases due to spike-induced calcium influx to the presynaptic
terminal, after which
- (ii) a fraction :math:`u` of available resources is consumed to produce the
post-synaptic current.
Between spikes, :math:`u` decays back to zero with time constant :math:`\tau_f`
and :math:`x` recovers to 1 with time constant :math:`\tau_d`.
In summary, the dynamics of STP is given by
.. math::
\begin{aligned}
\frac{du}{dt} & = -\frac{u}{\tau_f}+U(1-u^-)\delta(t-t_{sp}),\nonumber \\
\frac{dx}{dt} & = \frac{1-x}{\tau_d}-u^+x^-\delta(t-t_{sp}), \\
\frac{dI}{dt} & = -\frac{I}{\tau_s} + Au^+x^-\delta(t-t_{sp}),
\end{aligned}
where :math:`t_{sp}` denotes the spike time and :math:`U` is the increment
of :math:`u` produced by a spike. :math:`u^-, x^-` are the corresponding
variables just before the arrival of the spike, and :math:`u^+`
refers to the moment just after the spike. The synaptic current generated
at the synapse by the spike arriving at :math:`t_{sp}` is then given by
.. math::
\Delta I(t_{spike}) = Au^+x^-
where :math:`A` denotes the response amplitude that would be produced
by total release of all the neurotransmitter (:math:`u=x=1`), called
absolute synaptic efficacy of the connections.
**Model Examples**
- `STP for Working Memory Capacity <https://brainpy-examples.readthedocs.io/en/latest/working_memory/Mi_2017_working_memory_capacity.html>`_
**STD**
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = bp.dyn.LIF(1)
>>> neu2 = bp.dyn.LIF(1)
>>> syn1 = bp.dyn.STP(neu1, neu2, bp.connect.All2All(), U=0.2, tau_d=150., tau_f=2.)
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 28.)], monitors=['syn.I', 'syn.u', 'syn.x'])
>>> runner.run(150.)
>>>
>>>
>>> # plot
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 7)
>>>
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.u'][:, 0], label='u')
>>> plt.plot(runner.mon.ts, runner.mon['syn.x'][:, 0], label='x')
>>> plt.legend()
>>>
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.I'][:, 0], label='I')
>>> plt.legend()
>>>
>>> plt.xlabel('Time (ms)')
>>> plt.show()
**STF**
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import matplotlib.pyplot as plt
>>>
>>> neu1 = bp.dyn.LIF(1)
>>> neu2 = bp.dyn.LIF(1)
>>> syn1 = bp.dyn.STP(neu1, neu2, bp.connect.All2All(), U=0.1, tau_d=10, tau_f=100.)
>>> net = bp.dyn.Network(pre=neu1, syn=syn1, post=neu2)
>>>
>>> runner = bp.dyn.DSRunner(net, inputs=[('pre.input', 28.)], monitors=['syn.I', 'syn.u', 'syn.x'])
>>> runner.run(150.)
>>>
>>>
>>> # plot
>>> fig, gs = bp.visualize.get_figure(2, 1, 3, 7)
>>>
>>> fig.add_subplot(gs[0, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.u'][:, 0], label='u')
>>> plt.plot(runner.mon.ts, runner.mon['syn.x'][:, 0], label='x')
>>> plt.legend()
>>>
>>> fig.add_subplot(gs[1, 0])
>>> plt.plot(runner.mon.ts, runner.mon['syn.I'][:, 0], label='I')
>>> plt.legend()
>>>
>>> plt.xlabel('Time (ms)')
>>> plt.show()
**Model Parameters**
============= ============== ======== ===========================================
**Parameter** **Init Value** **Unit** **Explanation**
------------- -------------- -------- -------------------------------------------
tau_d 200 ms Time constant of short-term depression.
tau_f 1500 ms Time constant of short-term facilitation.
U .15 \ The increment of :math:`u` produced by a spike.
A 1 \ The response amplitude that would be produced by total release of all the neurotransmitter
delay 0 ms The decay time of the current :math:`I` output onto the post-synaptic neuron groups.
============= ============== ======== ===========================================
**Model Variables**
=============== ================== =====================================================================
**Member name** **Initial values** **Explanation**
--------------- ------------------ ---------------------------------------------------------------------
u 0 Release probability of the neurotransmitters.
x 1 A Normalized variable denoting the fraction of remain neurotransmitters.
I 0 Synapse current output onto the post-synaptic neurons.
=============== ================== =====================================================================
**References**
.. [1] Stevens, Charles F., and Yanyan Wang. "Facilitation and depression
at single central synapses." Neuron 14, no. 4 (1995): 795-802.
.. [2] Abbott, Larry F., J. A. Varela, Kamal Sen, and S. B. Nelson. "Synaptic
depression and cortical gain control." Science 275, no. 5297 (1997): 221-224.
.. [3] Abbott, L. F., and Wade G. Regehr. "Synaptic computation."
Nature 431, no. 7010 (2004): 796-803.
.. [4] Tsodyks, Misha, Klaus Pawelzik, and Henry Markram. "Neural networks
with dynamic synapses." Neural computation 10.4 (1998): 821-835.
.. [5] Tsodyks, Misha, and Si Wu. "Short-term synaptic plasticity."
Scholarpedia 8, no. 10 (2013): 3153.
"""
[docs] def __init__(
self,
pre: NeuGroup,
post: NeuGroup,
conn: Union[TwoEndConnector, Tensor, Dict[str, Tensor]],
U: float = 0.15,
tau_f: float = 1500.,
tau_d: float = 200.,
tau: float = 8.,
A: float = 1.,
delay_step: Union[int, Tensor, Initializer, Callable] = None,
method: str = 'exp_auto',
name: str = None
):
super(STP, self).__init__(pre=pre, post=post, conn=conn, name=name)
self.check_post_attrs('input')
self.check_pre_attrs('spike')
# parameters
self.tau_d = tau_d
self.tau_f = tau_f
self.tau = tau
self.U = U
self.A = A
# connections
self.pre_ids, self.post_ids = self.conn.require('pre_ids', 'post_ids')
# variables
self.num = len(self.pre_ids)
self.x = bm.Variable(bm.ones(self.num, dtype=bm.float_))
self.u = bm.Variable(bm.zeros(self.num, dtype=bm.float_))
self.I = bm.Variable(bm.zeros(self.num, dtype=bm.float_))
self.delay_type, self.delay_step, self.delay_I = init_delay(delay_step, self.I)
# integral
self.integral = odeint(method=method, f=self.derivative)
def reset(self):
self.x.value = bm.zeros(self.num)
self.u.value = bm.zeros(self.num)
self.I.value = bm.zeros(self.num)
self.delay_I.reset(self.I)
@property
def derivative(self):
dI = lambda I, t: -I / self.tau
du = lambda u, t: - u / self.tau_f
dx = lambda x, t: (1 - x) / self.tau_d
return JointEq([dI, du, dx])
def update(self, t, dt):
# delayed pre-synaptic spikes
if self.delay_type == 'homo':
delayed_I = self.delay_I(self.delay_step)
elif self.delay_type == 'heter':
delayed_I = self.delay_I(self.delay_step, bm.arange(self.pre.num))
else:
delayed_I = self.I
self.post.input += bm.syn2post(delayed_I, self.post_ids, self.post.num)
self.I.value, u, x = self.integral(self.I, self.u, self.x, t, dt=dt)
syn_sps = bm.pre2syn(self.pre.spike, self.pre_ids)
u = bm.where(syn_sps, u + self.U * (1 - self.u), u)
x = bm.where(syn_sps, x - u * self.x, x)
self.I.value = bm.where(syn_sps, self.I + self.A * u * self.x, self.I)
self.u.value = u
self.x.value = x
if self.delay_type in ['homo', 'heter']:
self.delay_I.update(self.I)
