1. Fully connected network. Assume a fully connected network of Poisson neurons with firing rate . 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 to a postsynaptic neuron is, it generates a postsynaptic current
where is the moment when the presynaptic neuron fired a spike and is the synaptic time constant.
a) Assume that each neuron in the network fires at the same rate . Calculate the mean and the variance of the input current to neuron .
Hint: Use the methods of Chapter 8
b) Assume that all weights of equal weight . Show that the mean input to neuron is independent of and that the variance decreases with .
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 A.
2. Stochastically connected network. Consider a network analogous to that discussed in the previous exercise, but with a synaptic coupling current
which contains both an excitatory and an inhibitory component.
a) Calculate the mean synaptic current and its variance assuming arbitrary coupling weights . How do mean and variance depend upon the number of neurons ?
b) Assume that the weights have a value . How do the mean and variance of the synaptic input current scale as a function of ?
3. Mean-field model. Consider a network of neurons with all-to-all connectivity and scaled synaptic weights . The transfer function (rate as a function of input potential) is piecewise linear.
The rate vanishes for and is constant (in units of the maximal rate) for .
The dynamics of the input potential of a neuron are
(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
Inhibitory neurons have a firing rate
Assume that we have a large network of excitatory and inhibitory neurons, where . The input to an excitatory neuron is
where is the rate of excitatory neuron and the rate of inhibitory neuron . The input to an inhibitory neuron is
(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 for the excitatory population activity; express the activity of the inhibitory population by 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 and the external input .
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