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Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics.

Puelma Touzel M, Wolf F - PLoS Comput. Biol. (2015)

Bottom Line: While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects.We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak.The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

View Article: PubMed Central - PubMed

Affiliation: Department for Nonlinear Dynamics, Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany.

ABSTRACT
The response of a neuronal population over a space of inputs depends on the intrinsic properties of its constituent neurons. Two main modes of single neuron dynamics-integration and resonance-have been distinguished. While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects. To understand better how a resonator's frequency preference emerges from its intrinsic dynamics and contributes to its local area's population firing rate dynamics, we analyze the dynamic gain of an analytically solvable two-degree of freedom neuron model. In the Fokker-Planck approach, the dynamic gain is intractable. The alternative Gauss-Rice approach lifts the resetting of the voltage after a spike. This allows us to derive a complete expression for the dynamic gain of a resonator neuron model in terms of a cascade of filters on the input. We find six distinct response types and use them to fully characterize the routes to resonance across all values of the relevant timescales. We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak. We determine the parameter regions for the existence of an intrinsic frequency and for subthreshold and spiking resonance, finding all possible intersections of the three. The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

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Related in: MedlinePlus

From input to ensemble response: numerics and prediction.Model output for the default parameter set: τI = 1ms, σI = 1, τV = 10ms, θ = 1, τw = 20ms, fint = 20Hz (g = 3.15). Left: in in the case of an oscillation of amplitude A = 0.05 and input frequency ω = (2π)20 rad/s. Right: in the case of a step of height A = 0.1. The example realization shown is the one with the maximum number of spikes from the sample ensemble. The red line is the response calculated using the analytical expressions for the oscillation and step response, Eqs (41) and (63), respectively.
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pcbi.1004636.g002: From input to ensemble response: numerics and prediction.Model output for the default parameter set: τI = 1ms, σI = 1, τV = 10ms, θ = 1, τw = 20ms, fint = 20Hz (g = 3.15). Left: in in the case of an oscillation of amplitude A = 0.05 and input frequency ω = (2π)20 rad/s. Right: in the case of a step of height A = 0.1. The example realization shown is the one with the maximum number of spikes from the sample ensemble. The red line is the response calculated using the analytical expressions for the oscillation and step response, Eqs (41) and (63), respectively.

Mentions: To illustrate the dynamic ensemble response, we show in Fig 2 an example of input, intrinsic, and output variable time series produced by the model for two choices of signal in the mean channel, : a weak oscillation of amplitude A and frequency ω and, separately, a step of height Δ. In addition, we show the corresponding population firing rate dynamics obtained from a histogram of the spike times of the sample ensemble produced by the two inputs. Code to produce this plot can be found in the supplemental material. The input modulation structures the spike times produced by the ensemble relative to the stationary response in a way that only becomes salient at this population level. We motivate consideration of the analytical expressions for the linear response function (Eq (41)) and the step response function (Eq (63)), obtained later in this paper, by plotting their curves, which accurately overlie the profile of the two respective measured histograms. While the input oscillation produces modulation in the output spiking at only one frequency, the step input produces a response that has power across a broad band of frequencies.


Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics.

Puelma Touzel M, Wolf F - PLoS Comput. Biol. (2015)

From input to ensemble response: numerics and prediction.Model output for the default parameter set: τI = 1ms, σI = 1, τV = 10ms, θ = 1, τw = 20ms, fint = 20Hz (g = 3.15). Left: in in the case of an oscillation of amplitude A = 0.05 and input frequency ω = (2π)20 rad/s. Right: in the case of a step of height A = 0.1. The example realization shown is the one with the maximum number of spikes from the sample ensemble. The red line is the response calculated using the analytical expressions for the oscillation and step response, Eqs (41) and (63), respectively.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4697854&req=5

pcbi.1004636.g002: From input to ensemble response: numerics and prediction.Model output for the default parameter set: τI = 1ms, σI = 1, τV = 10ms, θ = 1, τw = 20ms, fint = 20Hz (g = 3.15). Left: in in the case of an oscillation of amplitude A = 0.05 and input frequency ω = (2π)20 rad/s. Right: in the case of a step of height A = 0.1. The example realization shown is the one with the maximum number of spikes from the sample ensemble. The red line is the response calculated using the analytical expressions for the oscillation and step response, Eqs (41) and (63), respectively.
Mentions: To illustrate the dynamic ensemble response, we show in Fig 2 an example of input, intrinsic, and output variable time series produced by the model for two choices of signal in the mean channel, : a weak oscillation of amplitude A and frequency ω and, separately, a step of height Δ. In addition, we show the corresponding population firing rate dynamics obtained from a histogram of the spike times of the sample ensemble produced by the two inputs. Code to produce this plot can be found in the supplemental material. The input modulation structures the spike times produced by the ensemble relative to the stationary response in a way that only becomes salient at this population level. We motivate consideration of the analytical expressions for the linear response function (Eq (41)) and the step response function (Eq (63)), obtained later in this paper, by plotting their curves, which accurately overlie the profile of the two respective measured histograms. While the input oscillation produces modulation in the output spiking at only one frequency, the step input produces a response that has power across a broad band of frequencies.

Bottom Line: While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects.We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak.The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

View Article: PubMed Central - PubMed

Affiliation: Department for Nonlinear Dynamics, Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany.

ABSTRACT
The response of a neuronal population over a space of inputs depends on the intrinsic properties of its constituent neurons. Two main modes of single neuron dynamics-integration and resonance-have been distinguished. While resonator cell types exist in a variety of brain areas, few models incorporate this feature and fewer have investigated its effects. To understand better how a resonator's frequency preference emerges from its intrinsic dynamics and contributes to its local area's population firing rate dynamics, we analyze the dynamic gain of an analytically solvable two-degree of freedom neuron model. In the Fokker-Planck approach, the dynamic gain is intractable. The alternative Gauss-Rice approach lifts the resetting of the voltage after a spike. This allows us to derive a complete expression for the dynamic gain of a resonator neuron model in terms of a cascade of filters on the input. We find six distinct response types and use them to fully characterize the routes to resonance across all values of the relevant timescales. We find that resonance arises primarily due to slow adaptation with an intrinsic frequency acting to sharpen and adjust the location of the resonant peak. We determine the parameter regions for the existence of an intrinsic frequency and for subthreshold and spiking resonance, finding all possible intersections of the three. The expressions and analysis presented here provide an account of how intrinsic neuron dynamics shape dynamic population response properties and can facilitate the construction of an exact theory of correlations and stability of population activity in networks containing populations of resonator neurons.

Show MeSH
Related in: MedlinePlus