In the previous section, the formalism of membrane potential densities has been applied to a single population driven by spikes arriving at some rate at the synapse of type , where . In a population that is coupled to itself, the spikes that drive a given neuron are, at least partially, generated within the same population. For a homogeneous population with self-coupling the feedback is then proportional to the population activity ; cf. Chapter 12.
We now formulate the interaction between several coupled populations using the Fokker-Planck equation for the membrane potential density (Section 13.4.1) and apply it subsequently to the special cases of a population of excitatory integrate-and-fire neurons interacting with a population of inhibitory ones (Section 13.4.2).
We consider multiple populations . The population with index contains neurons and its activity is denoted by . We recall that is the total number of spikes emitted by population in a short interval .
Let us suppose that population sends its spikes to another population . If each neuron in population receives input from all neurons in the total spike arrival rate from population is therefore (‘full connectivity’). If each neuron in population receives connections only from a subset of randomly chosen neurons of population , then the total spike arrival rate is (’random connectivity’ with a connection probability from population to population ). We assume that all connections from to have the same weight .
For each population we write down a Fokker-Planck equation analogous to Eq. (13.16). Neurons are leaky integrate-and-fire neurons. Within a population , all neurons have the same parameters , , , and in particular the same firing threshold . For population the Fokker-Planck equation for the evolution of membrane potential densities is then
The population activity is the flux through the threshold (cf. Eq. (13.20)), which gives in our case
Thus populations interact with each other via the variable ; cf. Fig. 13.6.
In Chapter 12 we have already discussed the stationary state of the population activity in a population coupled to itself. With the methods discussed in Chapter 12 we were, however, unable to study the stability of the stationary state; nor were we able to make any prediction about potential time-dependent solutions.
We now give a more complete characterization of a network of leaky integrate-and-fire neurons consisting of two populations: a population of excitatory neurons coupled to a population with inhibitory neurons (79). The structure of the network is shown in Fig. 13.6B. Each neuron (be it excitatory or inhibitory) receives connections from the excitatory population each one with weight ; it also receives connections from the inhibitory population () and furthermore connections from an external population (weight ) with neurons that fire at a fixed rate . Each spike causes, after a delay of a voltage jump of mV and the threshold is 20mV above resting potential. Note that each neuron receives four times as many excitatory than inhibitory inputs so that the total amount of inhibition balances excitation if inhibition is four times stronger (), but the relative strength is kept as a free parameter. Also note that, in contrast to Chapter 12, the weight here has units of voltage and directly gives the amplitude of the voltage jump: .
The population can be in a state of asynchronous irregular activity (AI), where neurons in the population fire at different times (’asynchronous’ firing) and the distribution of interspike intervals is fairly broad (’irregular’ firing of individual neurons); cf. Fig. 13.7B. This is the state that corresponds to the considerations of stationary activity in Chapter 12. However, with a slight change of parameters, the same network can also be in a state of fast synchronous regular (SR) oscillations. It is characterized by periodic oscillations of the population activity and a sharply peaked interval distribution of individual neurons. The network can also be in the state of synchronous irregular firing (SI) either with fast (SI fast) or with slow (SI slow) oscillations of the population activity. The oscillatory temporal structure emerges despite the fact that the input has a constant spike arrival rate. It can be traced back to an instability of the asynchronous firing regime toward oscillatory activity.
With the mathematical approach of the Fokker-Planck equations, it is possible to determine the instabilities analytically. The mathematical approach will be presented in Section 13.5.2, but Fig. 13.7A already shows the result. Stability of a stationary state of asynchronous firing is most easily achieved in the regime where inhibition dominates excitation. Since each neuron receives four times as many excitatory as inhibitory inputs, inhibition must be at least four times as strong () as excitation. In order to get non-zero activity despite the strong inhibition, the external input alone must be sufficient to make the neurons fire. ’Input=1’ corresponds to an average external input just sufficient to reach the firing threshold, without additional input from the network.
Consider now the regime of strong inhibition () and strong input, say spikes from the external source arrive at a rate leading to a mean input of amplitude 4. To understand how the network can run into an instability, let us consider the following intuitive argument. Suppose a momentary fluctuation leads to an increase in the total amount of activity in the excitatory population. This causes, after a transmission delay an increase in inhibition and, after a further delay a suppression of excitation. If this feedback loop is strong enough, an oscillation with period may appear, leading to fast-frequency (’SI fast’) oscillations (79).
If the external input is, on its own, not sufficient to keep the network going (’input ’), then a similar small fluctuation may eventually turn the network from a self-activated state into a quiescent state where the only activity is that caused by external input. It then needs another fluctuation (caused by variations of spike arrivals from the external source) to kick it back into a short burst of activity (79). This leads to slow irregular, but synchronous bursts of firing (’SI slow’).
Finally, if inhibition is globally weak compared to excitation (), then the population is in a state of high activity where each neuron fires close to its maximal rate (set by the inverse of an absolute refractory period). This high-activity state is unstable compared to fast, but regular oscillations (’SR’). Typically, the population can split up into two or more subgroups, the number of which depends on the transmission delay (79; 181).
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