The gain function of rate models can always be chosen such that, for constant input , the population activity in the stationary state of asynchronous firing is correctly described. The dynamic equations that describe the approach to the stationary state in a rate model are, however, to a certain degree ad hoc. This means that the analysis of transients as well as the stability analysis in recurrent networks will, in general, give different results in rate models than in spiking neuron models.
In Section 15.2 we have seen that spiking neuron models with a large amount of escape noise exhibit a population activity that follows the input potential . In this case, it is therefore reasonable to define a rate model in which the momentary activity reflects the momentary input potential . An example is Eq. (15.1), which corresponds to a ’quasi-stationary’ treatment of the population activity, because the transform is identical to the stationary gain function, except for a change in the units of the argument, as discussed in the text after Eq. (15.1). A similar argument can also be made for exponential integrate-and-fire neurons with diffusive noise, as we will see later in this section.
Eq. (15.13) makes explicit that the population activity in the rate model reflects a low-pass filtered version of the input current. The transient response to a step in the input current is therefore slow.
Since differential equations are more convenient than integrals, we rewrite the input potential in the form of Eq. (15.3) which we repeat here for convenience
The input potential resulting from the integration of Eq. (15.14) is to be inserted into the function to arrive at the population activity . Note that the input current in Eq. (15.14) can arise from external sources, from other populations or from recurrent activity in the network itself.
Let us consider a network consisting of populations. Each population contains a homogeneous population of neurons. The input into population arising from other populations and from recurrent coupling within the population is described as
Here is the activity of population and is the number of presynaptic neurons in population that are connected to a typical neuron in population ; the time course and strength of synaptic connections are described by and , respectively; see Chapter 12.
We describe the dynamics of the input potential of population with the differential equation (15.14) and use for each population the quasi-stationary rate model where is the gain function of the neurons in population . The final result is
Eq. (15.19) is the starting point for some of the models in Part IV of this book.
To improve the description of transients, we start from Eq. (15.13), but insert an arbitrary filter ,
Eq. (15.21) is called the Linear-Nonlinear-Poisson (LNP) model (94; 478). It is also called a cascade model because it can be interpreted as a sequence of three processing steps. First, input is filtered with an arbitrary linear filter , which yields the input potential . Second, the result is passed through a nonlinearity . Third, in case of a single neurons, spikes are generated by an inhomogeneous Poisson process with rate . Since, in our model of a homogeneous population, we have many similar neurons, we drop the third step and interpret the rate directly as the population activity.
For we are back at Eq. (15.13). The question arises whether we can make a better choice of the filter than a simple low-pass with the membrane time constant . In Chapter 11 it is shown how an optimal filter can be determined experimentally by reverse correlation techniques.
Here we are interested in deriving the optimal filter from the complete population dynamics. The LNP model in Eq. (15.21) is an approximation of the population dynamics that is more correctly described by the Fokker-Planck equations in Ch. 13 or by the integral equation of time-dependent renewal theory in Ch. 14. Let us recall that in both approaches, we can linearize the population equations around a stationary state of asynchronous firing which is obtained with a mean input at some noise level . The linearization of the population dynamics about yields a filter . We use this filter, and arrive at a variant of Eq. (15.21)
where is a scaled version of the frequency-current curve and a constant which matches the slope of the gain function at the reference point for the linearization (373). Models with this, or similar choices of , describe transient peaks in the population activity surprisingly well; see e.g., (214; 33; 373).
Rate models based on Eq. (15.22) can also be used to describe coupled populations. Stability of a stationary state is correctly described by Eq. (15.22), if the filter in the argument on the right-hand-side reflects the linearization of the full population dynamics around , but not if the filter is derived by linearization around some other value of the activity.
So far we have focused on the initial transient after a step in the input current. After the initial transient, however, follows a second, much slower phase of adaptation during which the population response decreases, even if the stimulation is kept constant. For single neurons, adaptation has been discussed in Chapter 6.
In a population of neurons, adaptation can be described as an effective decrease in the input potential. If a population of non-adaptive neurons has an activity described by the gain function , then the population rate model for adaptive neurons is
where describes the amount of adaptation that neurons have accumulated
where is the asymptotic level of adaptation that is attained if the population continuously fires at a constant rate . The asymptotic level is approached with a time constant . Eqs. (15.25) and (15.26) are a simplified version of the phenomenological model proposed in Benda and Herz (48).
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