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12. Neuronal Populations

Identical Neurons: A Mathematical Abstraction

Definition: In a population of \(N\) neurons, we define the population activity

\[ A(t)=\frac{1}{N}\sum_{j=1}^{N}\sum_f\delta(t-t_j^f) \]

Homogeneous Networks

By the term "homogeneous" we mean:

all neurons \(1\leq i\leq N\) are identical;
all neurons receive the same external input \(I_i^{\t{ext}}(t)=I^{\t{ext}}(t)\);
the interaction strength (weight) \(w_{ij}\) is statistically uniform, \(w_{ij}\approx w_0\).

Take homogeneous population of integrate-and-fire model as an example; In a simple leaky case, the dynamics are:

\[ \tau_m\frac{\rmd u_i}{\rmd t}=-u_i+RI_i(t) \]

We assume that all the neurons connect to each other (including itself) with the same coupling strength \(w_{ij}=w_0\), and each input spike generates a post-synaptic current with some generic time course \(\alpha(t-t_j^f)\); then the \(I_i\) is:

\[ \begin{aligned} I_i(t)=&\sum_{j=1}^{N}\sum_f w_{ij}\alpha(t-t_j^f)+I^{\t{ext}}(t)\\ =& w_0N\left(\frac{1}{N}\sum_{j=1}^{N}\sum_f\alpha(t-t_j^f)\right)+I^{\t{ext}}(t)\\ =& w_0N\int_{0}^{+\infty}\alpha(s)A(t-s)\rmd s+I^{\t{ext}}(t) \end{aligned} \]

From the result we can see that \(I_i(t)\) is independent of index \(i\), which corresponds to the homogeneity. And it takes an elegant convolution form, with two parameters \(\alpha(t)\) and population activity \(A(t)\).

Heterogeneous Networks

What if not all neurons are identical, or the input received is not the same? That is, what happens to a heterogeneous networks? The instant motivation is to divide them into groups, and find a typical parameter to describe the consistent properties in a certain group. Let us firstly suppose the input \(I\) is identical for all the neurons, but certain parameter \(\theta\) varies slightly between individual neurons.