CN C4 Spiking Neuron Models

Spike Train, 脉冲序列

Spike, 脉冲

神经元的输出信号由多个短促的电脉冲（Electric Pulse）构成。

• 电脉冲：动作电位（Action Potential）或脉冲（Spike）。

一个神经元的AP具有相似的形式:

$A_U=100\ mV$

$\Delta T=1-2\ \rm ms$。

脉冲的形式不包含信息，脉冲的数量(number)和脉冲时刻（timing）包含信息。

Spike Train, 脉冲序列

脉冲是神经元信息传递的基本单元。

脉冲序列（Spike Train）:一个神经元发放的一系列脉冲。

• 脉冲序列中的任意脉冲之间都有一定的时间间隔。
• 不应期（Refractory Period）：两个脉冲之间的最短间隔。在不应期内接收到信号，神经元很难或不能发放新的脉冲。

数学描述：神经动力学

$$S_i(t) = \sum_{f}{\delta(t - t _i^{(f)})}$$

Where $f$ is the Dirac function:

Delta Function

The delta function is a generalized function that can be defined as the limit of a class of delta sequences.

The delta function, sometimes called Dirac’s delta function or the impulse symbol, takes the feature:

• $\forall x \ne 0,\delta(x) = 0$
• $\int_{-\infty}^{\infty}\delta(x){\rm d} x = 1$

Matlab Fundamentals

Basic arithmetic and function

arithmetic op	example	ans
/	22 / 11	2
^	2^4	16
e	12e-3	0.0120
sin	sin(pi/2)	1
exp	exp(1)	2.7183
log	log(2.7183)	1.0000
round	round(1.6)	2
abs	abs(-8)	8

Matrix and Array

a = [1 2 3;4 5 6]             
b = zeros(2)
c = a(:,1)
d = rand(2,3)
index = find(a(:,1)>2) # index = 2

$$a = \left[\begin{array}{ccc}1& 2& 3 \\4 &5& 6\end{array}\right]$$

$$b = \left[\begin{array}{cc}0& 0\\0 &0\end{array}\right]$$

$$c = \left[\begin{array}{c}1 \\4\end{array}\right]$$

$$d = \left[\begin{array}{ccc}0.4057 &0.6729& 0.7143\\ 0.8890& 0.2041& 0.8486\end{array}\right]$$

Plot

figure; 
timeWindow = 100; 
neuronNum = 4; 
pixel = rand(1, neuronNum); 
spikesNum = floor(pixel*timeWindow); 

hold on
colors = ['r', 'g', 'b', 'k'] 
for i=1:neuronNum
	spikeTrains = floor(rand(1,spikesNum(i))*timeWindow); 
	for j=1:length(spikeTrains) 
		line([spikeTrains(j) spikeTrains(j)], [0 1]+i-1, 'color', colors(mod(i,4)+1)) 
	end
end 
hold off 

xlabel('Time (ms)'); 
ylabel('Neuron No'); 
ylim([0 neuronNum+1])
set(gca,'ytick',0:1:neuronNum+1)

conclusion:

• figure
• line
• plot
• hold on
• xlabel
• ylabel
• ylim
• set

Modeling neurons

Neurons modeling is to abstract a computable mathmetical model, neuron model, from biological neurons.

RECALL Membrane potential is a difference of electrical potential across the membrane, which is the fundaments of the ability of information processing of neurons.

Idealized neurons

• Dendrites as Input component, receives signals and produces graded local signals.

• Soma(胞体) as center process unit, performs the important nonlinear processing.

• Axon as Output component, propagates signals to other neurons.

McCulloch-Pitts Neuron Model

• Two neurons connect through axon-cell body;
• One neuron provides inputs to the other through the connection;
• A certain amount of inputs is needed for a neuron to fire.
• Fire/not fire a spike, no half spikes or in-between spikes
• Perform logic calculation.

$$g(x_1,x_2,\cdots,x_n) = g(\vec{x}) = \sum_{i = 1}^{n}x_i$$

$$y = f(g(x)) = \begin{cases}1,\ g(x) \ge \theta\\0,\ otherwise\end{cases}$$

Simplified Neuron Model

Synapse is the connection component of axon-dendrite or axon-soma, which is the key structure of signal propagates.

Complex Cell (多室神经元)

LNP model

• L, Linear Filter, 线性滤波器: 瞬时发放速率如何响应输入的电流脉冲
• N, Nonlinearity: 滤波器输出的非线性(Nonlinearity)组合
• P, Poisson Spike Generator, 泊松脉冲产生器

Nernst potential, 内伦斯特电位

细胞膜是良好的绝缘体，细胞内外浓度分别为$n_1, n_2$, 电位分别为$u(x_1), u(x_2)$。

$$\frac{n_1}{n_2} = \exp[-\frac{qu(x_1)-qu(x_2)}{kT}] = \exp[-\frac{q\Delta u}{kT}]$$

$$\Delta u = -\frac{kT}{q}\ln\frac{n_1}{n_2} = \frac{kT}{q}\ln\frac{n_2}{n_1}$$

Neuronal Dynamics

Leaky Integrate-and-Fire Model,LIF

与HH模型的区别在于LIF神经元把所有跨膜电阻成分看做一个整体， 并假设它是不变的，这样就将模型大大化简了。

$$\tau_m\frac{\rm d \it u}{\rm d \it t} = RI(t) - (u(t) - u_{rest})$$

电流守恒：

$$I(t) = I_R + I_C$$

其中电阻电流$I_R$

$$I_R = \frac{u_R}{R} = \frac{u(t) - u_{rest}}{R}$$

其中$u_R,u(t),u_{rest}$分别是穿过电阻的电压，$t$时刻膜电压和静息电压。

$I_C$是对电容$C$充电的电流，

$$I_C = \frac{\rm d \it q}{\rm d \it t} = \frac{\rm d \it Cu}{\rm d \it t} = C\frac{\rm d \it u}{\rm d \it t}$$

综上可有

$$I(t) =\frac{u(t) - u_{rest}}{R} + C\frac{\rm d \it u}{\rm d \it t}$$

形式变换有

$$RI(t) =u(t) - u_{rest} + RC\frac{\rm d \it u}{\rm d \it t}$$

令$\tau_m = RC$，

$$\tau_m\frac{\rm d \it u}{\rm d \it t} = RI(t) - (u(t) - u_{rest})$$

Constant Current Input

假定输入电流为常数$I$，即$\forall t,I(t) = I$，

$$\begin{align} \tau_m\frac{\rm d \it u}{\rm d \it t} = RI - (u(t) - u_{rest})\\ \frac{\rm d \it u}{\rm d \it t} = - \frac{u(t)}{\tau_m} + \frac{RI}{\tau_m} + \frac{u_{rest}}{\tau_m} \end{align}$$

若 $- \frac{u}{\tau_m} + \frac{RI}{\tau_m} + \frac{u_{rest}}{\tau_m} = 0$，则 $u = u_{rest} + RI$为一个解。

若 $- \frac{u}{\tau_m} + \frac{RI}{\tau_m} + \frac{u_{rest}}{\tau_m} \ne 0$​，形式变换有

$$\frac{\rm d \it u}{(u - RI - u_{rest})} = - \frac{\rm d\it t}{\tau_m}$$

左右两侧同时不定积分有

$$\begin{align} \int\frac{\rm d \it u}{(u - RI - u_{rest})} - \int(- \frac{\rm d\it t}{\tau_m}) = 0\\ \ln(u - RI - u_{rest}) + \frac{t}{\tau_m} + c = 0\\ u(t) = c'e^{-\frac{t}{\tau_m}} + RI + u_{rest} \end{align}$$

$c, c'$ is constant.

$t = 0$时，$u(0) = c' + RI + u_{rest}$，故 $c' = u(0) - RI - u_{rest}$，

$$u(t) = (u(0) - RI - u_{rest})e^{-\frac{t}{\tau_m}} + RI + u_{rest}$$

假设$u(0) = u_{rest} = 0$，

$$u(t) = RI(1-e^{-\frac{t}{\tau_m}}) = RI(1-e^{-\frac{t}{RC}})$$

The time period

$$T = \tau_m \ln\frac{RI}{RI - V_{th}} + \Delta_{abs}$$

General time-varying Current Input

给定

$$u(t_0) = u_{rest}$$

$$u(t) = u_{rest}\exp(-\frac{t-t_0}{\tau_m}) + \frac{R}{\tau_m}\int_0^{t - t_0}\exp{(-\frac{s}{\tau_m})I(t - s)\rm d\it s}$$

cycle input current

Afferent Inputs

postsynaptic potential, PSP, 突触后电位: ；突触后神经元对突触前AP的响应。递质的释放依靠突触前神经去极化和Ca+进入突触前末梢，递质释放后通过突触间隙cleft 扩散到突触后膜，并与后膜上的特殊受体结合，改变后膜对离子的通透性(打开离子通道Ion Channel)，使后膜电位发生变化。

• 兴奋型： EPSP, excitatory postsynaptic potential
• 抑制型：IPSP, inhibitory postsynaptic potential

Synaptic Transmission

• Chemical Synapse：参照PSP的形成过程，电信号转变为化学信号转变为电信号。
• Electrical Synapse：神经元之间通过特定膜蛋白直接进行电连接，可能和神经元间的的同步synchronization相关。

突触和脉冲

• 空间位置与作用：不同空间位置突触作用不同；
• 空间位置与影响： 远端突触的影响要小于直接连接到胞体的突触的影响；
• 计算行为：饱和、加、分流抑制。

Computational Model

PSP

$$u_i(t) - u_{rest} =: \epsilon_{ij}(t)$$

• EPSP: $\epsilon_{ij}(t) > 0$
• IPSP: $\epsilon_{ij}(t) \le 0$

AP

$$u_i(t) = \sum_j\sum_f\epsilon_{ij}(t-t_j^{(f)}) + u_{rest}$$

$t_j^{(f)}$ is the moment of neuron $j$ firing spike.

Firing threshold

If $u_i(t)$ reaches threshold $v$ from below, neuron $𝑖$ fires a spike.

Nonlinear Neuroal Dynamics

• hyperpolarization, 超极化：神经元发放一个AP后，膜电压并不是直接降低到静息电位 ，而是降到一个更低的值。
• 促使神经元发放的膜电压通常是大于静息电位20-30mV，需要大量的 突触前脉冲在短时间内到达（20-50 spikes）。

Spike Response Model, SRM

$$u_i(t) = \eta(t - \hat{t}_i) + \sum_j\sum_f\epsilon_{ij}(t-t_j^{(f)}) + u_{rest}$$

$$\eta(t - t_i^{(f)}) = \begin{cases}\frac{1}{\Delta t},\ 0<t - t_i^{(f)} < \Delta t\\ -\eta_0\exp(-\frac{t - t_i^{(f)}}{\tau}),\ \Delta t < t - t_i^{(f)} \end{cases}$$