Stochastic Processes in the Neurosciences

Tuckwell, Henry C.

doi:10.1137/1.9781611970159

Cited by 245 publications

(199 citation statements)

References 0 publications

Supporting

Mentioning

196

Contrasting

Unclassified

Order By: Relevance

“…The most direct, and probably best-explored, view of the IF model is as a linear diffusion process (Ricciardi, 1977;Tuckwell, 1989;Burkitt, 2006). This connection allows us to apply powerful tools from stochastic calculus (Karatzas and Shreve, 1997) to understand the behavior of this model.…”

Section: The If Model As a Diffusion Processmentioning

confidence: 99%

“…However, the stochastic calculus leads directly to partial differential equation (Risken, 1996;Paninski et al, 2004c) or integral equation (Siegert, 1951;Buoncore et al, 1987;Plesser and Tanaka, 1997;Burkitt and Clark, 1999;DiNardo et al, 2001;Paninski et al, 2007a;Mullowney and Iyengar, 2007) methods for computing the likelihood. For example, it is well-known (Tuckwell, 1989) that the probability density of the next spike satisfies…”

Section: The If Model As a Diffusion Processmentioning

confidence: 99%

See 1 more Smart Citation

Statistical Models of Spike Trains

Paninski¹,

Brown²,

Iyengar³

2009

Stochastic Methods in Neuroscience

View full text Add to dashboard Cite

Section: The If Model As a Diffusion Processmentioning

confidence: 99%

Section: The If Model As a Diffusion Processmentioning

confidence: 99%

Statistical Models of Spike Trains

Paninski¹,

Brown²,

Iyengar³

2009

Stochastic Methods in Neuroscience

View full text Add to dashboard Cite

“…The constant decay assumed here and represented by the term d is thus a key simplification. As general references on stochastic processes and on the computation of first passage times the reader may consult, for example, the books by Ricciardi (1977), Tuckwell (1989), or Risken (1996; see also Berg (1993) for applications to other problems in biology.…”

Section: Appendixmentioning

confidence: 99%

Impact of Correlated Synaptic Input on Output Firing Rate and Variability in Simple Neuronal Models

2000

View full text Add to dashboard Cite

Cortical neurons are typically driven by thousands of synaptic inputs. The arrival of a spike from one input may or may not be correlated with the arrival of other spikes from different inputs. How does this interdependence alter the probability that the postsynaptic neuron will fire? We constructed a simple random walk model in which the membrane potential of a target neuron fluctuates stochastically, driven by excitatory and inhibitory spikes arriving at random times. An analytic expression was derived for the mean output firing rate as a function of the firing rates and pairwise correlations of the inputs. This stochastic model made three quantitative predictions. (1) Correlations between pairs of excitatory or inhibitory inputs increase the fluctuations in synaptic drive, whereas correlations between excitatory-inhibitory pairs decrease them. (2) When excitation and inhibition are fully balanced (the mean net synaptic drive is zero), firing is caused by the fluctuations only. (3) In the balanced case, firing is irregular. These theoretical predictions were in excellent agreement with simulations of an integrate-and-fire neuron that included multiple conductances and received hundreds of synaptic inputs. The results show that, in the balanced regime, weak correlations caused by signals shared among inputs may have a multiplicative effect on the input-output rate curve of a postsynaptic neuron, i.e. they may regulate its gain; in the unbalanced regime, correlations may increase firing probability mainly around threshold, when output rate is low; and in all cases correlations are expected to increase the variability of the output spike train. Key words: random-walk; integrate-and-fire; computer simulation; spike synchrony; oscillations; cross-correlation; balanced inhibition; cerebral cortexThe output of a typical cortical neuron depends on the activity of a large number of synaptic inputs-several thousands of them, as estimated by anatomical techniques (White, 1989; Braitenberg and Shüz, 1997). What kind of response should be expected from a postsynaptic neuron driven by so many inputs? Answering this question in detail requires a deep understanding of dendritic integration, synaptic function, and spike generation mechanisms; but, given the large numbers commonly involved, as a first approximation it is natural to cast the problem in statistical terms. The strategy then is to compute the output responses of a model neuron (or their statistics), given a set of driving inputs with known statistical properties. These inputs may be either independent or temporally correlated. In the latter case, spikes from different input neurons arrive close together in time more often or less often than expected by chance.In general, the situation with independent inputs is easier to analyze, and for many applications it is probably a good approximation. However, there are at least three reasons why the effects of correlations on single cells should be fully characterized. First, correlations in spike counts have indeed been o...

show abstract

“…It still provides challenging problems and this is despite a very large number of works have been devoted to this objective [1][2][3][4]. The theoretical description of single neuron dynamics has been attempted in the framework of different models, such as the classical Hodgkin-Huxley, the FitzHugh-Nagumo and the leaky integrate-and-fire model.…”

Section: Introductionmentioning

confidence: 99%

“…In the present work we will consider an integrate-and-fire model with a cubic nonlinearity as was suggested by Tuckwell [1] driven by a subthreshold periodic signal with a long period compared to the characteristic relaxation time towards the resting voltage. The small random contri- bution to the dynamics of the membrane voltage is represented by Gaussian white noise, which only rarely drives the voltage to the threshold.…”

Section: Introductionmentioning

confidence: 99%

Neuron firing in driven nonlinear integrate-and-fire models

Kostur

Schindler

Talkner

et al. 2007

Mathematical Biosciences

View full text Add to dashboard Cite

Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.

show abstract

Stochastic Processes in the Neurosciences

Cited by 245 publications

References 0 publications

Statistical Models of Spike Trains

Statistical Models of Spike Trains

Impact of Correlated Synaptic Input on Output Firing Rate and Variability in Simple Neuronal Models

Neuron firing in driven nonlinear integrate-and-fire models

Contact Info

Product

Resources

About