The interspike-interval statistics of non-renewal neuron models

The spontaneous activity of a neuron as well as its response to constant stimuli is often characterized by irregular and irreproducible spike trains. This variability is due to various sources of neuronal noise. Many statistical features of these spike trains such as the variability of interspike intervals (ISIs) and spike counts have been theoretically understood by means of reduced stochastic neuron models that are amenable to a mathematical analysis. Most of these models belong to the class of renewal models, in which the ISIs within a spike train are statistically independent. Because of this convenient mathematical property, renewal models have proven very successful in explaining the first-order ISI statistics. However, numerous experimental studies have reported correlations between ISIs – a higher-order statistics that cannot be described by a renewal model. The present thesis deals with the theoretical analysis of non-renewal models. In particular, two scenarios are investigated more closely: neurons in the presence of colored noise representing temporally correlated inputs and neurons exhibiting spike-frequency adaptation in the presence of different sources of noise. Using extended integrate-and-fire neuron models, approximation techniques are put forward to analytically calculate the stationary ISI statistics in the suband suprathreshold spiking regime. The analysis focuses on the serial correlation coefficient (SCC). The perfect integrate-and-fire (PIF) model, which is the canonical model for suprathreshold periodic firing, serves as the basis for further extensions that render the model non-renewal. In the first part, the PIF model is driven by colored noise in the form of a multi-dimensional Ornstein-Uhlenbeck process, which represents a broad class of possible colored noise processes. For this case, general formulas for the coefficient of variation, the SCC, the ISI density and the Fano factor are derived. For the special case of exponentially-correlated noise, it is shown that colored noise with a large correlation time can lead to strongly peaked ISI densities with a large skewness and kurtosis. As another special case, a superposition of harmonic noise and exponentially-correlated noise is considered. With such colored noise, complex serial correlation structures that have been observed in electro-sensory neurons of paddlefish can be explained theoretically. In the second part, a white-noise driven PIF model is extended by a deterministic adaptation current that realizes an inhibitory feedback. An expression for the SCC is derived for weak noise, which is valid for arbitrary feedback strengths and adaptation time constants. The theory reproduces short-range, negative serial correlations, which have been frequently reported for neurons with spike-frequency adaptation. It also reveals an optimal ratio of time scales for which adjacent ISIs are most strongly anti-correlated. Furthermore, two dynamical regimes are identified, where the SCC is either negative at all lags or alternates in sign. By applying weak-noise approximation techniques, the stationary Fokker-Planck equation corresponding to the twodimensional Langevin equation can be solved, which results in explicit expressions for the membrane potential distribution and the distribution of the adaptation variable upon firing. Finally, it is shown how the ISI density, the linear response function and the power spectrum can be computed for slow adaptation currents and arbitrary noise strength. The theory explicitly reveals the inverse Gaussian statistics of ISIs, the high-pass filter property of adaptation and the reduction of the power spectrum at low frequencies due to negative serial correlations. In the third part, the effect of a stochastic adaptation current is studied. This situation arises when the adaptation current is mediated by a finite population of “adaptation channels” (channel noise). By using the diffusion approximation it is shown that slow adaptation channels introduce an intrinsic source of colored noise. The station-