Nonlinear signal processing by noisy spiking neurons

Neurons are excitable cells which communicate with each other via electrical signals. In general, these signals are processed by the Neurons in a nonlinear fashion, the exact mathematical description of which is still an open problem in neuroscience. In this thesis, the broad topic of nonlinear signal processing is approached from two directions. The first part of the thesis is devoted to the question how input signals modulate the neural response. The second part of the thesis is concerned with the nonlinear reconstruction of input signals from the neural output and with the estimation of the amount of the transmitted information. The nonlinear modulation of the neural response is studied by extending the linear response theory. For the analytically tractable leaky integrate-and-fire model, we derive the second-order response function that allows to predict the time-dependent firing rate in response to general time-dependent signals, which is not only a singleneuron statistics which is often measured in experiments, but can also be regarded as the instantaneous population rate of a group of uncoupled neurons subject to the same external drive. The weakly nonlinear theory reveals for the leaky integrateand-fire model several interesting features. For cosine signals, the theory successfully describes the excitation of higher harmonics, a frequency-dependent change of the time-averaged firing rate, and a strong nonlinear interaction of multiple input signals. The weakly nonlinear regime emerges as a transition between the regime of linear encoding and a highly nonlinear regime in which the neuron exhibits mode-locking behavior. The transition emerges not only for increasing signal amplitude, as suggested by the Volterra expansion of the firing rate, but also for decreasing strength of the intrinsic noise. Beyond the analytically tractable leaky integrate-and-fire model with white Gaussian current noise, we also measure numerically the second-order response function for a biophysically more realistic spiking neuron model with channel noise. The similarity of the weakly nonlinear response for the two models suggests that the discussed features do not hinge on the particularities of the models but are of a more general nature. The nonlinear reconstruction of input signals and the nonlinear estimation of the information content of the neural spike count is studied by extending the linear lower bound for the mutual information. For the special case of a static input signal, we derive a quadratic lower bound which can be used as a simple estimate for the mutual information. Our result is expressed in terms of linear and nonlinear correlation coefficients which can be computed for any signal-processing system with time-independent input and output. From a decoding perspective, the quadratic lower bound corresponds to the minimal information that is recovered from the output via a quadratic reconstruction. We demonstrate that the quadratic signal reconstruction outperforms the linear theory if the mean of the output variable exhibits a nonlinear signal dependence. More interestingly, however, the quadratic lower bound can be used as an estimate of the mutual information for systems which exhibit a signal-dependent variance and for which the linear lower bound is always zero. For two different neuron models we demonstrate that a quadratic reconstruction of the input signal can outperform the linear reconstruction significantly. This finding implies that higher-order correlations between signal and spike count carry information which is not accounted for by the linear correlations. For the estimation of the mutual information between signal and spike count we find that the linear and