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A complex-valued firing-rate model that approximates the dynamics of spiking networks.

Schaffer ES, Ostojic S, Abbott LF - PLoS Comput. Biol. (2013)

Bottom Line: These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization.We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models.Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

View Article: PubMed Central - PubMed

Affiliation: Department of Neuroscience, Department of Physiology and Cellular Biophysics, Columbia University College of Physicians and Surgeons, New York, New York, United States of America.

ABSTRACT
Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

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Comparison of the linear response of the complex-valued rate model and integrate-and-fire models.A–B. Gain and C–D. phase of the linear response of the QIF (red), EIF (green), LIF (blue) and complex-valued rate (black) models. A. and C. Baseline coefficient of variation of 0.1. B. and D. Baseline coefficient of variation of 0.7. Insets in A and C show gain and phase, respectively, of response near the resonant frequency of 50 Hz. In all cases, the baseline firing rate was 50 Hz.
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pcbi-1003301-g004: Comparison of the linear response of the complex-valued rate model and integrate-and-fire models.A–B. Gain and C–D. phase of the linear response of the QIF (red), EIF (green), LIF (blue) and complex-valued rate (black) models. A. and C. Baseline coefficient of variation of 0.1. B. and D. Baseline coefficient of variation of 0.7. Insets in A and C show gain and phase, respectively, of response near the resonant frequency of 50 Hz. In all cases, the baseline firing rate was 50 Hz.

Mentions: The linear response of the complex-valued rate model is given by (Methods) and , where(6)The results shown in Figure 4 are based on using the steady-state rate of the EIF model to compute , but the results are quite insensitive to which neuron model is used to define .


A complex-valued firing-rate model that approximates the dynamics of spiking networks.

Schaffer ES, Ostojic S, Abbott LF - PLoS Comput. Biol. (2013)

Comparison of the linear response of the complex-valued rate model and integrate-and-fire models.A–B. Gain and C–D. phase of the linear response of the QIF (red), EIF (green), LIF (blue) and complex-valued rate (black) models. A. and C. Baseline coefficient of variation of 0.1. B. and D. Baseline coefficient of variation of 0.7. Insets in A and C show gain and phase, respectively, of response near the resonant frequency of 50 Hz. In all cases, the baseline firing rate was 50 Hz.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3814717&req=5

pcbi-1003301-g004: Comparison of the linear response of the complex-valued rate model and integrate-and-fire models.A–B. Gain and C–D. phase of the linear response of the QIF (red), EIF (green), LIF (blue) and complex-valued rate (black) models. A. and C. Baseline coefficient of variation of 0.1. B. and D. Baseline coefficient of variation of 0.7. Insets in A and C show gain and phase, respectively, of response near the resonant frequency of 50 Hz. In all cases, the baseline firing rate was 50 Hz.
Mentions: The linear response of the complex-valued rate model is given by (Methods) and , where(6)The results shown in Figure 4 are based on using the steady-state rate of the EIF model to compute , but the results are quite insensitive to which neuron model is used to define .

Bottom Line: These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization.We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models.Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

View Article: PubMed Central - PubMed

Affiliation: Department of Neuroscience, Department of Physiology and Cellular Biophysics, Columbia University College of Physicians and Surgeons, New York, New York, United States of America.

ABSTRACT
Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons.

Show MeSH