We focus on a Spike Response Model with escape noise; cf. Eqs. (9.1) - (9.3). If the firing rate is low, so that the interspike interval is much longer than the decay time of the refractory kernel , then we can truncate the sum over past firing times and keep track only of the effect of the most recent spike (182)
where denotes the last firing time .
Eq. (9.19) is called the ‘short-term memory’ approximation of the SRM and abbreviated as SRM. This model can be efficiently fitted to neural data (251) and will play an important role in Chapter 14 of Part III of this book. In order to emphasize that the value of the membrane potential depends only on the most recent spike, we write in the following instead of . Let us summarize the total effect of the input by introducing the ‘input potential’
which allows us to rewrite Eq. (9.19) as
The escape rate
depends on the time since the last spike and, implicitly, on the stimulating current . Hence is similar to the hazard variable of stationary renewal theory. The arguments of Chapter 7 can be generalized to the case of time-dependent input which gives rise to a time-dependent input potential . Given that the neuron has fired its last spike at time and that we know the input for we can calculate the probability density that the next spike occurs at time
Compared to the standard stationary renewal theory discussed in Chapter 7, there are two important differences. First, the Spike Response Model with escape noise provides a direct path from stationary to time-dependent renewal theory. Second, interval distributions can be linked to refractoriness and vice versa. More precisely, a reduced firing intensity immediately after a spike is an indication that the distance between the membrane potential and the threshold is increased. The reason can be either a hyperpolarizing spike after-potential or an increase in the firing threshold immediately after a spike.
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