Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.

/pdf/spiking-neuron-models-single-neurons-populations-plasticity-58gzrd6ih6.pdf

Spiking Neuron Models: Single Neurons, Populations, Plasticity

Tables of integrals

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

The authors evaluated 4 sequential sampling models for 2-choice decisions--the Wiener diffusion, Ornstein-Uhlenbeck (OU) diffusion, accumulator, and Poisson counter models--by fitting them to the response time (RT) distributions and accuracy data from 3 experiments. Each of the models was augmented with assumptions of variability across trials in the rate of accumulation of evidence from stimuli, the values of response criteria, and the value of base RT across trials. Although there was substantial model mimicry, empirical conditions were identified under which the models make discriminably different predictions. The best accounts of the data were provided by the Wiener diffusion model, the OU model with small-to-moderate decay, and the accumulator model with long-tailed (exponential) distributions of criteria, although the last was unable to produce error RTs shorter than correct RTs. The relationship between these models and 3 recent, neurally inspired models was also examined.

A comparison of sequential sampling models for two-choice reaction time

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker–Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin–Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to – the irregular nature of spiking in cortical neurons and neural gain modulation.

https://link.springer.com/content/pdf/10.1007%2Fs00422-006-0068-6.pdf

A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling

For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.

On the M / M /1 Queue with Catastrophes and Its Continuous Approximation

A stochastic model in tumor growth.

The algorithm given by Buonocore et al. (1) to evaluate first-passage-time p.d.f.'s for Wiener and Ornstein-Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion proc- esses. Several examples are thoroughly discussed along with some computational results. equation was proposed to determine first-passage-time p.d.f.'s through time- dependent boundaries for time-homogeneous one-dimensional diffusion processes with natural boundaries. This equation was seen to be particularly manageable for computational purposes in the cases of Wiener and Ornstein-Uhlenbeck processes. Indeed, for these the kernel of the integral equation can be made continuous, thus overcoming some crucial difficulties arising from the singular nature of the integral equations for the first-passage-time p.d.f. Such continuity was made possible by a suitable choice of two arbitrary time-dependent functions that appear in the kernel of the integral equations. Here, in view of the relevance of first-passage-time problems in numerous applied fields (see, for instance, Buonocore et al. (1) and references therein), we intend to provide an extension of the results of Buonocore et al. (1) in two directions: on the one hand, we shall prove that the method for regularizing the kernel of the integral equation is valid not only for the Wiener and the Ornstein-Uhlenbeck processes but can also be used for a more general class of diffusion processes whose free transition

On the evaluation of first-passage-time probability densities via non-singular integral equations

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

Virginia Giorno

Papers

An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling

On the M / M /1 Queue with Catastrophes and Its Continuous Approximation

A stochastic model in tumor growth.

On the evaluation of first-passage-time probability densities via non-singular integral equations

Some remarks on the Rayleigh process