For the simple integrate-and-fire model the voltage V is given as a solution of the equation:
C\frac{dV}{dt}=I.
This is just the derivate of the law of capacitance Q=CV. When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold V_{\text{th}}, at which point a delta function spike occurs.
A shortcoming of the simple integrate-and-fire model is that it implements no time-dependent memory. If the model receives a below-threshold signal at some time, it will retain that voltage boost until it fires again. This characteristic is not in line with observed neuronal behavior.
In the leaky integrate-and-fire model, the memory problem is solved by adding a "leak" term \frac{-1}{R}V (R is the resistance and \tau=RC) to the membrane potential:
\frac{dV}{dt}=\frac{-1}{\tau}V+\frac{1}{C}I.
This reflects the diffusion of ions that occurs through the membrane when some equilibrium is not reached in the cell.
To solve :math:numref:`membrane` we start by looking at a simpler differential equation:
\frac{df}{dt}=af\text{, where } f:\mathbb{R}\to\mathbb{R} \text{ and } a\in\mathbb{R}.
Here the solution is given by f(t)=e^{at}.
When you add another function g to the right hand side of our linear differential equation,
\frac{df}{dt}=af+g
this is now a non-homogeneous differential equation. Things (can) become more complicated.
This kind of differential equation is usually solved with "variation of constants" which gives us the following solution:
f(t)=e^{ct}\int_{0}^t g(s)e^{-cs}ds.
This is obviously not a particularly handy solution since calculating the integral in every step is very costly.
With exact integration, these costly computations can be avoided.
But only for certain functions g! I.e. if g satisfies (is a solution of):
\left(\frac{d}{dt}\right)^n g= \sum_{i=1}^{n}a_i\left(\frac{d}{dt}\right)^{i-1} g
for some n\in \mathbb{N} and a sequence (a_i)_{i\in\mathbb{N}}\subset \mathbb{R}.
For example this would be the case for g=\frac{e}{\tau_{syn}}t e^{-t/\tau_{\text{syn}}} (an alpha funciton), where \tau_{\text{syn}} is the rise time.
The non-homogeneous differential equation is reformulated as a multidimensional homogeneous linear differential equation:
\frac{d}{dt}y=Ay
where
A=\begin{pmatrix} a_{n} & a_{n-1} & \cdots & \cdots & a_1 & 0 \\ 1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 & 0 \\ 0 & 0 & \ddots & 1 & 0 & 0 \\ 0 & 0 & \cdots & 0 & \frac{1}{C} & -\frac{1}{\tau} \\ \end{pmatrix}
by choosing y_1,...,y_n canonically as:
\begin{align*} y_1 &= \left(\frac{d}{dt}\right)^{n-1}g\\ \vdots &= \vdots\\ y_{n-1} &= \frac{d}{dt}g\\ y_{n} &= g\\ y_{n+1} &= f. \end{align*}
This makes ist very easy to determine the solution as
y(t)= e^{At}y_0
and
y_{t+h}=y(t+h)=e^{A(t+h)}\cdot y_0=e^{Ah}\cdot e^{At}\cdot y_0=e^{Ah}\cdot y_t.
This means that once we have calculated A, propagation consists of multiplications only.
The dynamics of the membrane potential V is given by:
\frac{dV}{dt}=\frac{-1}{\tau}V+\frac{1}{C}I
where \tau is the membrane time constant and C is the capacitance. I is the sum of the synaptic currents and any external input:
Postsynaptic currents are alpha-shaped, i.e. the time course of the synaptic current \iota due to one incoming spike is
\iota (t)= \frac{e}{\tau_{syn}}t e^{-t/\tau_{\text{syn}}}.
The total input I to the neuron at a certain time t is the sum of all incoming spikes at all grid points in time t_i\le t plus an additional piecewise constant external input I_{\text{ext}}:
I(t)=\sum_{i\in\mathbb{N}, t_i\le t }\sum_{k\in S_{t_i}}\hat{\iota}_k \frac{e}{\tau_{\text{syn}}}(t-t_i) e^{-(t-t_i)/\tau_{\text{syn}}}+I_{\text{ext}}
S_t is the set of indices that deliver a spike to the neuron at time t, \tau_{\text{syn}} is the rise time and \iota_k represents the "weight" of synapse k.
First we make the substitutions:
\begin{align*} y_1 &= \frac{d}{dt}\iota+\frac{1}{\tau_{syn}}\iota \\ y_2 &= \iota \\ y_3 &= V \end{align*}
for the equation
\frac{dV}{dt}=\frac{-1}{Tau}V+\frac{1}{C}\iota
we get the homogeneous differential equation (for y=(y_1,y_2,y_3)^t)
\frac{d}{dt}y= Ay= \begin{pmatrix} \frac{1}{\tau_{syn}}& 0 & 0\\ 1 & \frac{1}{\tau_{syn}} & 0\\ 0 & \frac{1}{C} & -\frac {1}{\tau} \end{pmatrix} y.
The solution of this differential equation is given by y(t)=e^{At}y(0) and can be solved stepwise for a fixed time step h:
y_{t+h}=y(t+h)=e^{A(t+h)}y(0)=e^{Ah}e^{At}y(0)=e^{Ah}y(t)=e^{Ah}y_t.
The complete update for the neuron can be written as
y_{t+h}=e^{Ah}y_t + x_{t+h}
where
x_{t+h}+\begin{pmatrix}\frac{e}{\tau_{\text{syn}}}\\0\\0\end{pmatrix}\sum_{k\in S_{t+h}}\hat{\iota}_k
as the linearity of the system permits the initial conditions for all spikes arriving at a given grid point to be lumped together in the term x_{t+h}. S_{t+h} is the set of indices k\in 1,....,K of synapses that deliver a spike to the neuron at time t+h.
The matrix e^{Ah} in the C++ implementation of the model in NEST is constructed here.
Every matrix entry is calculated twice. For inhibitory postsynaptic inputs (with a time constant \tau_{syn_{in}}) and excitatory postsynaptic inputs (with a time constant \tau_{syn_{ex}}).
And the update is performed here. The first multiplication evolves the external input. The others are the multiplication of the matrix e^{Ah} with y. (For inhibitory and excitatory inputs)
[1] | RotterV S & Diesmann M (1999) Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biologial Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570 |