LIF模型

关键词

Leaky(泄露): 表示如果神经元输入只有一个，且不足以让膜电势超过阈值时，由于细胞膜不断进行膜内外离子交换，膜电势会自动发生泄露逐渐回落到静息状态；
Intergrate(集成): 表示神经元会接收所有与该神经元相连的轴突末端发送来的脉冲，并将所接收的所有脉冲信号进行集成(求和)；
Fired(激发): 表示当膜电势超过阈值时，神经元会发送脉冲，同时膜电位回落至静息电位Vreset。

模型

 解得

这里是任意常数， 控制指数下降速率， 越小 越快指数变化到,分析这个方程可以看到，初始时t=0时刻v=，其中取恰当的值就可以使等于脉冲神经元初始时刻电压，当t=∞时v=，该方程控制了电压v随时间指数稳定到平衡电压。

Python代码

import numpy as np
import matplotlib.pyplot as plt
 
fig = plt.figure(figsize=(5, 4))
ax = plt.subplot(111)
 
 
# Function that runs the simulation
# tau: time constant (in ms)
# t0, t1, t2: time of three input spikes
# w: input synapse weight
# threshold: threshold value to produce a spike
# reset: reset value after a spike
def LIF(tau=10, t0=20, t1=30, t2=35, w=0.8, threshold=1.0, reset=0.0):
    # Spike times, keep sorted because it's more efficient to pop the last value off the list
    times = [t0, t1, t2]
    times.sort(reverse=True)
    # set some default parameters
    duration = 100  # total time in ms
    dt = 0.1  # timestep in ms
    alpha = np.exp(-dt / tau)  # this is the factor by which V decays each time step
    V_rec = []  # list to record membrane potentials
    V = 0.0  # initial membrane potential
    T = np.arange(np.round(duration / dt)) * dt  # array of times
    spikes = []  # list to store spike times
    # run the simulation
    # plot everything (T is repeated because we record V twice per loop)
    ax.clear()
    for t in times:
        ax.axvline(t, ls=':', c='b')
 
    for t in T:
        V_rec.append(V)  # record
        V *= alpha  # integrate equations
        if times and t > times[-1]:  # if there has been an input spike
            V +=w
            times.pop()  # remove that spike from list
        V_rec.append(V)  # record V before the reset so we can see the spike
        if V > threshold:  # if there should be an output spike
            V = reset
            spikes.append(t)
    ax.plot(np.repeat(T, 2), V_rec, '-k', lw=2)
    for t in spikes:
        ax.axvline(t, ls='--', c='r')
    ax.axhline(threshold, ls='--', c='g')
    ax.set_xlim(0, duration)
    ax.set_ylim(-1, 2)
    ax.set_xlabel('Time (ms)')
    ax.set_ylabel('Voltage')
    plt.tight_layout()
    plt.show()
 
LIF()

绿色虚线表示发放阈值，黑色为神经元电压，蓝色虚线表示神经元接收到输入脉冲的时刻，红色虚线表示神经发放了一个脉冲。代码里设置神经元平衡电压为0，静息电位为0，发放阈值为1。