Exercises

1. Fully connected network. Assume a fully connected network of $N$ Poisson neurons with firing rate $\nu_{i}(t)=g(I_{i}(t))>0$. Each neuron sends its output spikes to all other neurons as well as back to itself. When a spike arrives at the synapse from a presynaptic neuron $j$ to a postsynaptic neuron $i$ is, it generates a postsynaptic current

 $I_{i}^{\rm syn}=w_{ij}\exp[-(t-t_{j}^{(f)})/\tau_{s}]\quad{\rm for}~{}~{}t>t_{% j}^{(f)}\,,$ (12.39)

where $t_{j}^{(f)}$ is the moment when the presynaptic neuron $j$ fired a spike and $\tau_{s}$ is the synaptic time constant.

a) Assume that each neuron in the network fires at the same rate $\nu$. Calculate the mean and the variance of the input current to neuron $i$.

Hint: Use the methods of Chapter 8

b) Assume that all weights of equal weight $w_{ij}=J_{0}/N$. Show that the mean input to neuron $i$ is independent of $N$ and that the variance decreases with $N$.

c) Evaluate mean and variance and the assumption that the neuron receives 4 000 inputs at a rate of 5Hz. The synaptic time constant is 5ms and $J_{0}=1\mu$A.

2. Stochastically connected network. Consider a network analogous to that discussed in the previous exercise, but with a synaptic coupling current

 $I_{i}^{\rm syn}=w_{ij}\left\{({1\over\tau_{1}})\exp[-(t-t_{j}^{(f)})/\tau_{1}]% -({1\over\tau_{2}})\exp[-(t-t_{j}^{(f)})/\tau_{2}]\right\}\quad{\rm for}~{}~{}% t>t_{j}^{(f)}\,,$ (12.40)

which contains both an excitatory and an inhibitory component.

a) Calculate the mean synaptic current and its variance assuming arbitrary coupling weights $w_{ij}$. How do mean and variance depend upon the number of neurons $N$?

b) Assume that the weights have a value $J_{0}/\sqrt{N}$. How do the mean and variance of the synaptic input current scale as a function of $N$?

3. Mean-field model. Consider a network of $N$ neurons with all-to-all connectivity and scaled synaptic weights $w_{ij}=J_{0}/N$. The transfer function (rate as a function of input potential) is piecewise linear.

 $f=g(h)={{h-h_{1}}\over{h_{2}-h_{1}}}\quad{\rm for~{}}h_{1}\leq h\leq h_{2}\,.$ (12.41)

The rate vanishes for $h and is constant $f=1$ (in units of the maximal rate) for $h>h_{2}$.

The dynamics of the input potential $h_{i}$ of a neuron $i$ are

 $\tau{dh_{i}\over dt}=-h_{i}+RI_{i}(t)\,,$ (12.42)

with

 $I(t)=I^{\rm ext}+\sum_{j}w_{ij}\alpha(t-t_{j}^{(f)})\,.$ (12.43)

(i) Find graphically the fixed points of the population activity in the network with connections as described above.

(ii) Determine the solutions analytically.

4. Mean-field in a Network of 2 populations

We study a network of excitatory and inhibitory neurons. Each excitatory neuron has, in the stationary state, a firing rate

 $e=f(I)=\gamma\,I\quad{\rm for~{}}I>0\quad{\rm and~{}else~{}}f(I)=0\,.$ (12.44)

Inhibitory neurons have a firing rate $s$

 $s=g(I)=I^{2}\quad{\rm for~{}}I>0\quad{\rm and~{}else~{}}g(I)=0\,.$ (12.45)

Assume that we have a large network of $N$ excitatory and $N$ inhibitory neurons, where $N\gg 1$. The input to an excitatory neuron $i$ is

 $I_{i}(t)=I_{0}+\sum_{k=1}^{N}{w\over N}e_{k}-\sum_{n=1}^{N}{1\over N}s_{n}$ (12.46)

where $e_{k}$ is the rate of excitatory neuron $k$ and $s_{n}$ the rate of inhibitory neuron $n$. The input to an inhibitory neuron $n$ is

 $I_{i}(t)=\sum_{k=1}^{N}{w\over N}e_{k}$ (12.47)

(i) Give the analytical solution for the steady state of the network. If there are several solutions, indicate stability of each of these.

[Hint: Introduce the parameter $A=\sum_{k=1}^{N}{1\over N}e_{k}$ for the excitatory population activity; express the activity of the inhibitory population by $A$ and insert the result into the excitatory equation]

(ii) Solve graphically for the stationary state of the activity in the network, for two qualitatively different regimes which you choose. Free parameters are the coupling strength $w$ and the external input $I$.