Abstract
Nerve cells in the brain generate sequences of action potentials with a complex statistics. Theoretical attempts to understand this statistics were largely limited to the case of a temporally uncorrelated input (Poissonian shot noise) from the neurons in the surrounding network. However, the stimulation from thousands of other neurons has various sorts of temporal structure. Firstly, input spike trains are temporally correlated because their firing rates can carry complex signals and because of cell-intrinsic properties like neural refractoriness, bursting, or adaptation. Secondly, at the connections between neurons, the synapses, usage-dependent changes in the synaptic weight (short-term plasticity) further shape the correlation structure of the effective input to the cell. From the theoretical side, it is poorly understood how these correlated stimuli, so-called colored noise, affect the spike train statistics. In particular, no standard method exists to solve the associated first-passage-time problem for the interspike-interval statistics with an arbitrarily colored noise. Assuming that input fluctuations are weaker than the mean neuronal drive, we derive simple formulas for the essential interspike-interval statistics for a canonical model of a tonically firing neuron subjected to arbitrarily correlated input from the network. We verify our theory by numerical simulations for three paradigmatic situations that lead to input correlations: (i) rate-coded naturalistic stimuli in presynaptic spike trains; (ii) presynaptic refractoriness or bursting; (iii) synaptic short-term plasticity. In all cases, we find severe effects on interval statistics. Our results provide a framework for the interpretation of firing statistics measured in vivo in the brain.
Highlights
The biophysics of action potential generation in a single nerve cell is well understood in the framework of the celebrated equivalent circuit model by Hodgkin and Huxley (Koch 1999)
We consider a perfect integrate-and-fire (PIF) neuron, which represents a reasonable approximation of neural spike generation in the mean-driven firing regime
Of the spike train with the post-synaptic current j (t) approximates the synaptic filtering by the conductance dynamics of synapse
Summary
The biophysics of action potential generation in a single nerve cell is well understood in the framework of the celebrated equivalent circuit model by Hodgkin and Huxley (Koch 1999). Theoretical studies often assume that input spike trains have no temporal structure but are completely random in time with a spiking statistics given by a Poisson process. This assumption has been successfully used to explain and self-consistently determine the irregular, Poisson-like firing patterns of cortical neurons and the emergence of network oscillations Sources that induce temporal correlations are signal-related processes (Baddeley et al 1997), neuronal refractoriness (Cateau and Reyes 2006) and bursting (Bair et al 1994), adaptation (Wang 1998) and network-generated oscillations (Buzsaki and Draguhn 2004). As a consequence of both the nonlinear neural dynamics and the temporally structured driving, the interspike-interval (ISI) statistics of the driven cell is complex (Bair et al 1994; Baddeley et al 1997; Compte et al 2003; Nawrot et al 2007)
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