Mathematical neuroscience: from neurons to circuits to systems

Abstract

Applications of mathematics and computational techniques to our understanding of neuronal systems are provided. Reduction of membrane models to simplified canonical models demonstrates how neuronal spike-time statistics follow from simple properties of neurons. Averaging over space allows one to derive a simple model for the whisker barrel circuit and use this to explain and suggest several experiments. Spatio-temporal pattern formation methods are applied to explain the patterns seen in the early stages of drug-induced visual hallucinations.

Neural interactions between flicker-induced self-organized visual hallucinations and physical stimuli

“…(ii) If so, can they interact with neural activity evoked by physical stimuli to affect perception? To tackle these questions we consider stereotyped geometric hallucinations, often triggered by migraine, drug intoxications, and empty-field flicker (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20); these phenomena are thought to arise from autonomous activity in visual cortex. Although little is known about how spontaneous activity affects the perception of physical stimuli, it is possible to approach the second question from another direction: How do physical stimuli affect perception of spontaneous activity?…”

“…1). We regard these hallucinations as spontaneous states because they are apparently self-organized by visual cortex (14)(15)(16)(17)(18)(19)(20) and can arise without visual stimulation; these perceptible states may be analogous to Kenet et al's (3) subliminal states (9). Their geometric forms are predicted by neural network models that generate stripe patterns of neural activity whose spatial orientation on cortex combines with the nonlinear retinocortical mapping to determine the perceptual effect (14)(15)(16)(17)(18)(19)(20) (Fig.…”

Robust Master-Slave Synchronization of Neuronal Systems

“…Mathematical modeling has made an enormous impact on neuroscience [1][2][3][4]35]. A variety of dynamic models of the electrical activity of neurons have been reported in the literature [2][3][4][43][44][45][46].…”

“…Neural engineers apply mathematical and computational models, electrical engineering, and signal processing of living neuronal tissues [1,2]. Two fundamental issues in neurosciences are the synchronization of individual neurons and the functional role of synchronized activity [3,4]. Synchronization of neuron's activities is necessary for memory, calculation, motion control, and diseases such as epilepsy [5][6][7][8].…”

“…Synchronized activity and temporal correlation are critical for encoding and exchanging information for neuronal information processing in the brain [2][3][4]. Synchronization approaches in neuronal systems are aimed at exploring the communication between neurons with the computing of coupling functions that resemble observed experimental electrical cell activity [6][7][8][9][10].…”