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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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The type of w-current depends on the values of the intrinsic parameters.(a) Intrinsic parameter phase diagram in (τw/τV, g). w can be depolarizing (g < 0) or hyperpolarizing (g > 0). w contributes an intrinsic frequency to the model in the colored region. The dynamics are unstable if g < −1. Iso-Ω lines are shown in white ( for large τw and  for small τw). (b) When , the phase diagram can be cast in (τw/τV, ΩτV)-space. Iso-g lines are shown in white. (See [29] for a similar plot).
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pcbi.1004636.g001: The type of w-current depends on the values of the intrinsic parameters.(a) Intrinsic parameter phase diagram in (τw/τV, g). w can be depolarizing (g < 0) or hyperpolarizing (g > 0). w contributes an intrinsic frequency to the model in the colored region. The dynamics are unstable if g < −1. Iso-Ω lines are shown in white ( for large τw and for small τw). (b) When , the phase diagram can be cast in (τw/τV, ΩτV)-space. Iso-g lines are shown in white. (See [29] for a similar plot).

Mentions: Here we present analysis of the phase diagram of the intrinsic dynamics of the model, which is a reparametrization of Fig.1 from [29]. Beyond that work, here we analyze the Ω-contour density and scaling behavior. For a fixed, constant value of Isyn, and with time in units of τV, the structure of the phase space of the single neuron dynamics described by Eq (1) is determined by a point in the τw/τVvs. g plane, the two parameters defining the intrinsic current, w (see Fig 1). For τw ≪ τV, w speeds up or slows down V depending on whether g is hyperpolarizing (g > 0) or depolarizing (g < 0) characterized by an effective time constantτeff=τV1+g.(2)While the dissipative voltage term stabilizes the voltage dynamics, the dynamics can be effectively unstable for g < −1, and we exclude this case. For depolarizing intrinsic current, there is a region where the two eigenvalues of the voltage solution, λ±, are complex and the model exhibits an intrinsic frequency, Ω = 2πfint, that varies asΩ=g-gcritτwτV(3)where gcrit = (τw − τV)2/4τwτV (see Methods for details). For a fixed g > 0, a given value of Ω can be achieved at both a high and a low value of τw. For fast τw, the Ω-contour density is high and the model exhibits high parameter sensitivity, while for large τw the contour density is low and the model is relatively insensitive to local parameter variation. Taking the respective limits, the set of isofrequency curves are linear for large τw with slope ∝ Ω2 and with a slope independent of Ω for small τw.


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

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

The type of w-current depends on the values of the intrinsic parameters.(a) Intrinsic parameter phase diagram in (τw/τV, g). w can be depolarizing (g < 0) or hyperpolarizing (g > 0). w contributes an intrinsic frequency to the model in the colored region. The dynamics are unstable if g < −1. Iso-Ω lines are shown in white ( for large τw and  for small τw). (b) When , the phase diagram can be cast in (τw/τV, ΩτV)-space. Iso-g lines are shown in white. (See [29] for a similar plot).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi.1004636.g001: The type of w-current depends on the values of the intrinsic parameters.(a) Intrinsic parameter phase diagram in (τw/τV, g). w can be depolarizing (g < 0) or hyperpolarizing (g > 0). w contributes an intrinsic frequency to the model in the colored region. The dynamics are unstable if g < −1. Iso-Ω lines are shown in white ( for large τw and for small τw). (b) When , the phase diagram can be cast in (τw/τV, ΩτV)-space. Iso-g lines are shown in white. (See [29] for a similar plot).
Mentions: Here we present analysis of the phase diagram of the intrinsic dynamics of the model, which is a reparametrization of Fig.1 from [29]. Beyond that work, here we analyze the Ω-contour density and scaling behavior. For a fixed, constant value of Isyn, and with time in units of τV, the structure of the phase space of the single neuron dynamics described by Eq (1) is determined by a point in the τw/τVvs. g plane, the two parameters defining the intrinsic current, w (see Fig 1). For τw ≪ τV, w speeds up or slows down V depending on whether g is hyperpolarizing (g > 0) or depolarizing (g < 0) characterized by an effective time constantτeff=τV1+g.(2)While the dissipative voltage term stabilizes the voltage dynamics, the dynamics can be effectively unstable for g < −1, and we exclude this case. For depolarizing intrinsic current, there is a region where the two eigenvalues of the voltage solution, λ±, are complex and the model exhibits an intrinsic frequency, Ω = 2πfint, that varies asΩ=g-gcritτwτV(3)where gcrit = (τw − τV)2/4τwτV (see Methods for details). For a fixed g > 0, a given value of Ω can be achieved at both a high and a low value of τw. For fast τw, the Ω-contour density is high and the model exhibits high parameter sensitivity, while for large τw the contour density is low and the model is relatively insensitive to local parameter variation. Taking the respective limits, the set of isofrequency curves are linear for large τw with slope ∝ Ω2 and with a slope independent of Ω for small τw.

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