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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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The parameter  as a function of firing rate and CV.A. Curves through  space along which values in B–D are evaluated. B. Imaginary part of  divided by . C. Real part of  versus CV. Black line corresponds to . D. Real part of  versus . B–D show  for the QIF (dotted lines), EIF (dashed lines), and LIF (solid lines), with the color indicating the corresponding line in  space shown in A.
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pcbi-1003301-g002: The parameter as a function of firing rate and CV.A. Curves through space along which values in B–D are evaluated. B. Imaginary part of divided by . C. Real part of versus CV. Black line corresponds to . D. Real part of versus . B–D show for the QIF (dotted lines), EIF (dashed lines), and LIF (solid lines), with the color indicating the corresponding line in space shown in A.

Mentions: The computations of the dominant nonzero eigenvalue of the Fokker-Planck operator for these models are described in the Methods, and the results are shown in Figure 2. Rather than expressing as a function of the input parameters and , we use an equivalent parameterization in terms of the output, expressing as a function of and the coefficient of variation (CV) of the spiking models. The space has a one-to-one mapping with the space [14], and working in this space allows us to plot results for all three neuron models on comparable axes. In Figure 2B–D, we show the imaginary and real parts of along the curves in the space of values depicted by the different colored traces in Figure 2A (these are curves of fixed for the exponential integrate-and-fire model, in particular  = 1, 2 and 4 mV).


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

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

The parameter  as a function of firing rate and CV.A. Curves through  space along which values in B–D are evaluated. B. Imaginary part of  divided by . C. Real part of  versus CV. Black line corresponds to . D. Real part of  versus . B–D show  for the QIF (dotted lines), EIF (dashed lines), and LIF (solid lines), with the color indicating the corresponding line in  space shown in A.
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Related In: Results  -  Collection

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

pcbi-1003301-g002: The parameter as a function of firing rate and CV.A. Curves through space along which values in B–D are evaluated. B. Imaginary part of divided by . C. Real part of versus CV. Black line corresponds to . D. Real part of versus . B–D show for the QIF (dotted lines), EIF (dashed lines), and LIF (solid lines), with the color indicating the corresponding line in space shown in A.
Mentions: The computations of the dominant nonzero eigenvalue of the Fokker-Planck operator for these models are described in the Methods, and the results are shown in Figure 2. Rather than expressing as a function of the input parameters and , we use an equivalent parameterization in terms of the output, expressing as a function of and the coefficient of variation (CV) of the spiking models. The space has a one-to-one mapping with the space [14], and working in this space allows us to plot results for all three neuron models on comparable axes. In Figure 2B–D, we show the imaginary and real parts of along the curves in the space of values depicted by the different colored traces in Figure 2A (these are curves of fixed for the exponential integrate-and-fire model, in particular  = 1, 2 and 4 mV).

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