12 Neuronal Populations


1. Fully connected network. Assume a fully connected network of NN Poisson neurons with firing rate νi(t)=g(Ii(t))>0\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 jj to a postsynaptic neuron ii is, it generates a postsynaptic current

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

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

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

Hint: Use the methods of Chapter 8

b) Assume that all weights of equal weight wij=J0/Nw_{ij}=J_{0}/N. Show that the mean input to neuron ii is independent of NN and that the variance decreases with NN.

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 J0=1μJ_{0}=1\muA.

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

Iisyn=wij{(1τ1)exp[-(t-tj(f))/τ1]-(1τ2)exp[-(t-tj(f))/τ2]}fort>tj(f),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 wijw_{ij}. How do mean and variance depend upon the number of neurons NN?

b) Assume that the weights have a value J0/NJ_{0}/\sqrt{N}. How do the mean and variance of the synaptic input current scale as a function of NN?

3. Mean-field model. Consider a network of NN neurons with all-to-all connectivity and scaled synaptic weights wij=J0/Nw_{ij}=J_{0}/N. The transfer function (rate as a function of input potential) is piecewise linear.

f=g(h)=h-h1h2-h1forh1hh2.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<h1h<h_{1} and is constant f=1f=1 (in units of the maximal rate) for h>h2h>h_{2}.

The dynamics of the input potential hih_{i} of a neuron ii are

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


I(t)=Iext+jwijα(t-tj(f)).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)=γIforI>0andelsef(I)=0 .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 ss

s=g(I)=I2forI>0andelseg(I)=0 .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 NN excitatory and NN inhibitory neurons, where N1N\gg 1. The input to an excitatory neuron ii is

Ii(t)=I0+k=1NwNek-n=1N1NsnI_{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 eke_{k} is the rate of excitatory neuron kk and sns_{n} the rate of inhibitory neuron nn. The input to an inhibitory neuron nn is

Ii(t)=k=1NwNekI_{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=k=1N1NekA=\sum_{k=1}^{N}{1\over N}e_{k} for the excitatory population activity; express the activity of the inhibitory population by AA 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 ww and the external input II.