What happens in our brain when we make a decision? What triggers a neuron to send out a signal? What is the neural code? This textbook for advanced undergraduate and beginning graduate students provides a thorough and up-to-date introduction to the fields of computational and theoretical neuroscience. It covers classical topics, including the Hodgkin-Huxley equations and Hopfield model, as well as modern developments in the field such as Generalized Linear Models and decision theory. Concepts are introduced using clear step-by-step explanations suitable for readers with only a basic knowledge of differential equations and probabilities, and are richly illustrated by figures and worked-out examples. End-of-chapter summaries and classroom-tested exercises make the book ideal for courses or for self-study. The authors also give pointers to the literature and an extensive bibliography, which will prove invaluable to readers interested in further study.
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For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations.
Neural signaling requires a large amount of metabolic energy 1 . Consequently, neurons are thought to communicate using efficient codes in which redundant information is discarded 2 . Theories of efficient coding 3 successfully predict several features of sensory systems. At early stages of visual processing, inputs coming from the external world are decorrelated in both space and time [4][5][6][7] ; through sensory adaptation 8 , codes are dynamically modified so as to maximize information transmission [9][10][11][12] ; and sensory adaptation on multiple timescales 11,13,14 could possibly reflect the statistics of the external world 15 .Sensory adaptation is at least partially a result of intrinsic properties of individual neurons and, in particular, of SFA. SFA is not only observed at the early stages of sensory processing, but is also widespread in cortical neurons embedded in highly recurrent networks. Often modeled by a single process with one specific timescale 16,17 , SFA also occurs on multiple timescales [18][19][20] . In pyramidal neurons of the rat somatosensory cortex, three or more processing steps away from sensory receptors, SFA is scale free 21 , meaning that the effective speed at which individual neurons adapt is not fixed but depends on the input. Scale-free adaptation can be captured by simple threshold models with a power law-decaying spike-triggered process 22 that possibly describes the combined action of Na + -channel inactivation [23][24][25] and ion channels mediating adaptation currents [26][27][28] .Thus, three questions arise. First, can the temporal features of spiketriggered currents and spike-triggered changes in firing threshold, possibly spanning multiple timescales, be directly extracted from experimental data? Second, can SFA be explained by these spiketriggered effects? And finally, do the timescales of SFA match the temporal statistics of the inputs received by individual neurons? If temporal characteristics of inputs and SFA were matched, SFA could lead to a perfect decorrelation of the information contained in one spike with that of the previous one of the same neuron, a phenomenon known as temporal whitening 29 . Temporal whitening in turn implies that, at a high signal-to-noise ratio (SNR), information transmission is enhanced 30 . RESULTSThe question of whether SFA is optimally designed for efficient coding can only be addressed if both the dynamics of SFA and the statistical properties of the inputs generated in biologically relevant situations are known. We used a combined theoretical and experimental approach to extract the dynamics of spike-triggered processes and SFA directly from in vitro recordings of cortical neurons. We then analyzed the synaptically driven membrane potential dynamics recorded in vivo from somatosensory neurons during active whisker sensation (data from ref. 31). Our overall goal was to study whether adaptation optimally removes the temporal correlations in the input to single neurons embedded in the highly recurrent network of the cortex.SFA i...
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