22.interacting populations and continuum models

interacting populations

Every heterogeneous network could be viewed as a combination of different homogeneous network, or the interaction among different populations. The simplest case is the balanced random network: with excitatory and inhibitory neurons.

Let’s look at a fully connected network with excitatory and inhibitory neurons. Excitatory neurons belong to population $n$ and inhibitory neurons belong to population $m$.

So the activity of population $n$ or population $m$ is:

$$ A_n(t)=\frac{1}{N_n}\sum_{j\in \Gamma_n}\sum\delta(t-t_j^{(f)}) $$

So for each single neuron in population $m$, which receives the inputs from all neurons in population $n$, the membrane potential is changing accordingly:

$$ \begin{align} h_i(t|\hat{t_i}) & = \sum_j\sum_fw_{ij}\eta(t-\hat{t_i},t-t_j^{(f)})\
& = \sum_mJ_{nm}\int_0^{\infty}\eta(t-\hat{t_i},s)\sum_{j\in \Gamma_m}\sum_f\frac{\delta(t-t_j^{f}-s)}{N_m}\
& = \sum_mJ_{nm}\int_0^{\infty}\eta(t-\hat{t},s)A_m(t-s)ds

\end{align} $$

That is just an example population with SRM0 neuron model. For a general population, the population activity evolution could be rewrite as:

$$ A_n(t) = \int_{-\infty}^t P_n(t|\hat{t})A_n(\hat{t})d\hat{t} $$

$P$ is the interval distribution, it could also be viewed as a kernel for population activity, which is determined by the post-synaptic potential (PSP).

Stationary states and gain function of interacting populations

When the network reaches stationary state, the activity is stable, so the inputs from one sub-population ($N$ for example) to another sub-population is more or less constant.

So the inputs could be viewed as $I_m$ (to population $M$), and the activity of this population is: $$ A_m=g_m(I_m), $$

with

$$ I_m = \sum_nJ_{mn}A_n, $$

we have

$$ A_m = g_m(\sum_n J_{mn}A_n) $$

spatial continuum network

Features: the auditory cortex is a good example, with neurons organized along an axis with continuously changing preferred frequency.

So the principle of analyzing this network is \emph{binning} the cortex with a binsize $d$. Each sub-population could be labeled according to its spatial location. For instance, population $N$ is the neurons between $[nd,(n+1)d]$, and in each bin, the neuron number is $N_n=\rho d$, where $\rho$ is the spatial density.

Then this network is a special case of interacting populations, with $d$ approaching zero.