Effects of noisy adaptation on neural spiking statistics Tilo Schwalger,Benjamin Lindner, Karin Fisch, Jan Benda Max-Planck Institute for the Physics of.

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Presentation transcript:

Effects of noisy adaptation on neural spiking statistics Tilo Schwalger,Benjamin Lindner, Karin Fisch, Jan Benda Max-Planck Institute for the Physics of Complex Systems, Dresden Stochastic Models in Neuroscience, Marseille 2010 LMU, Munich

Neurons show spike-frequency adaptation Gabbiani & Krapp, J Neurophysiol 2006 Intrinsic negative feedback

Outline What is the source of noise? ● Stratagy and introduction of the model ● ISI statistics for two cases i.“deterministic adaptation” + stochastic receptor current ii.“Stochastic adaptation” + deterministic receptor current ● “mixed case”

What is the source of variability? – The strategy ● Introduce variability by stochastic ion currents (channel noise), e.g. ● Modelling signal transduction of auditory receptor cells ● Receptor current ● Sodium current ● Potassium (delayed rectifier) current ● Adaptation current (M-type)

What is the source of variability? – The strategy ● Introduce variability by stochastic ion currents (channel noise), e.g. ● Modelling signal transduction of auditory receptor cells ● Receptor current ● Sodium current ● Potassium (delayed rectifier) current ● Adaptation current (M-type) ● Study separately the effect of a specific channel noise ● on the interspike interval (ISI) statistics ● ISI density, firing rate, coefficient of variation (CV), skewness, excess, … ● ISI serial correlation coefficient ● Compare different predictions with experimental data Interspike interval [ms ] Probability density

What is the source of variability? – The strategy ● Introduce variability by stochastic ion currents (channel noise), e.g. ● Modelling signal transduction of auditory receptor cells ● Receptor current ● Sodium current ● Potassium (delayed rectifier) current ● Adaptation current (M-type) ● Study separately the effect of a specific channel noise ● on the interspike interval (ISI) statistics ● ISI density, firing rate, coefficient of variation (CV), skewness, excess, … ● ISI serial correlation coefficient ● Compare different predictions with experimental data ● Here: stochastic adaptation current vs. Stochastic receptor current in a simplified model Interspike interval [ms ] Probability density

M-type adaptation current ● Two-state channel model ● (voltage-gated) ● Adaptation current with finite number of channels ● voltage-gated potassium current ● Slow kinetics # M-channels open # M- channels

Perfect integrate-and-fire model Stochastic adaption current White noise: fast receptor channels

Perfect integrate-and-fire model membrane Potential V M-channel activation fraction of open M-channels W Time [ms] Stochastic adaption current White noise: fast receptor channels threshold reset

Perfect integrate-and-fire model membrane Potential V M-channel activation fraction of open M-channels W Time [ms] Stochastic adaption current White noise: fast receptor channels threshold reset Firing rate Time [ms]

Diffusion approximation of channel noise ● adaptation split up in deterministic and noise part ● Additive noise approximation:

Two limit cases Deterministic adaptation

Two limit cases Stochastic adaptation Deterministic adaptation

Deterministic adaptation (N a →∞) ● Mean adaptation approximation reduced input current much larger than mean ISI

Deterministic adaptation (N a →∞) Inverse Gaussian ISI density Negative ISI correlations Gerstein & Mandelbrot, 1964

Stochastic adaptation current acts as a colored noise ● Slow modulation of instantaneous firing rate ● due to slow noise process η(t) ● average over fast dynamics of

Stochastic adaptation current acts as a colored noise ● Slow modulation of instantaneous firing rate ● due to slow noise process η(t) ● average over fast dynamics of → Colored noise-driven PIF neuron

Colored noise model captures ISI density

ISI density: B. Lindner, Phys Rev E 2004 small noise

Stochastic adaptation yields positive ISI correlations Positive ISI correlations B. Lindner, Phys Rev E 2004

ISI density – comparison Determistic adaptation Stochastic adaptation Same CV Inverse Gaussian fails!

ISI density – comparison Determistic adaptation Stochastic adaptation

How to discriminate from an inverse Gaussian density? skewnes s Inverse Gaussian

How to discriminate from an inverse Gaussian density? skewnes s excess (kurtosis) Inverse Gaussian

How to discriminate from an inverse Gaussian density? skewnes s excess (kurtosis) Inverse Gaussian Defin e inverse Gaussian distribution:

Separation of deterministic and stochastic adaptation Rescaled skewness Rescaled excess

Dependence on the adaptation time- scale Rescaled excess Serial correlation coefficient at lag 1

Skewness and excess of ISIs – theory ● PIF model driven by colored noise ● Fokker-Planck equation for p(v,η,t) ● Initial condition + fire&reset rule

Skewness and excess of ISIs – theory ● PIF model driven by colored noise ● Fokker-Planck equation for p(v,η,t) ● Initial condition ● ISI density + fire&reset rule

Skewness and excess of ISIs – theory ● PIF model driven by colored noise ● Fokker-Planck equation for p(v,η,t) ● Initial condition ● ISI density ● Variable transformation ● Fokker-Planck equation + fire&reset rule

Skewness and excess of ISIs – theory ● Small noise expansion

Mixed case – fast and slow fluctuations D>0 fixed, vary number of channels

Simulation of Hodgkin-Huxley type model with M-current Modified Traub-Miles model (Ermentrout, 2000) Mixed case Serial correlations

Summary ● Introduced an integrate-and-fire-model with stochastic adaptation current ● (channel noise) ● Case of deterministic adaptation current and white current noise: ● inverse Gaussian ISI distribution and negative serial correlations ● Case of stochastic adaptation current and no white noise: ● peaked ISI distribution and positive serial correlations ● Results might be useful to determine the dominant source of noise

Acknowledgements Karin Fisch LMU, Munich Benjamin Lindner MPIPKS, Dresden Jan Benda LMU, Munich

Variability depends on sound intensity Karin Fisch & Jan Benda, LMU