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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 6 distinct filter shapes in (νωL, ν∞)-space.(c, f) Region 1–6 denote the regions exhibiting qualitatively similar filter shapes. E.g. spiking resonance is by definition region 1 and 2. Not all of these six regions are accessible for a given QL. Colored lines (blue to red) represent the QL-dependent boundary below which filter shapes are forbidden because of unstable dynamics. We note that νωL, ν∞ → 0 = QL. An intrinsic frequency exists in region above the QL = 1/2 boundary. A voltage resonance exists in the region above the QL = 1 boundary. We show the accessible subset of corresponding filter shapes at representative positions within the regions (located at  and ) and at the border between regions (located at νωL, ν∞ = 10−1.5, 100, 101.5). (f) Same type of plot as (c), but for the phase response. π/2 and −π/2 are shown as top and bottom bounding dashed lines for the set of phase responses at each location. The gain and phase for the position denoted by the circle are shown in (a) and (c), and for the star in (b) and (e), respectively.
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pcbi.1004636.g009: The 6 distinct filter shapes in (νωL, ν∞)-space.(c, f) Region 1–6 denote the regions exhibiting qualitatively similar filter shapes. E.g. spiking resonance is by definition region 1 and 2. Not all of these six regions are accessible for a given QL. Colored lines (blue to red) represent the QL-dependent boundary below which filter shapes are forbidden because of unstable dynamics. We note that νωL, ν∞ → 0 = QL. An intrinsic frequency exists in region above the QL = 1/2 boundary. A voltage resonance exists in the region above the QL = 1 boundary. We show the accessible subset of corresponding filter shapes at representative positions within the regions (located at and ) and at the border between regions (located at νωL, ν∞ = 10−1.5, 100, 101.5). (f) Same type of plot as (c), but for the phase response. π/2 and −π/2 are shown as top and bottom bounding dashed lines for the set of phase responses at each location. The gain and phase for the position denoted by the circle are shown in (a) and (c), and for the star in (b) and (e), respectively.

Mentions: For this general case, we can introduce the relative quality factor for the full filter, νωL: = /ν1(ωL)//νlow. The response then depends on the five shape features, νlow, νhigh, ωL, QL, and νωL. Denoting ξ = τw/τc, so that and , we can re-express the response function asν1(ω)νlow=1+iν∞ξωωL1+iξν∞ωωL1-ω2/ωL2+iω/QLωL(46)with dynamic gainνm,1(ω)νlow=1+ν∞ξω2ωL21+ξν∞ω2ωL21-ω2ωL22+ω2QL2ωL2.(47)When ω = ωL, , which implicitly defines ξ in terms of νωL, QL and νhigh and closes the representation. Indeed, with time in units of and gain values relative to νlow, the shape of the filter depends only on this triplet: each of the six regions in (ν∞, νωL)-space defined by the boundaries ν∞ = 1, νωL = 1, and ν∞ = νωL provides filters of a qualitatively similar class (see Fig 9).


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

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

The 6 distinct filter shapes in (νωL, ν∞)-space.(c, f) Region 1–6 denote the regions exhibiting qualitatively similar filter shapes. E.g. spiking resonance is by definition region 1 and 2. Not all of these six regions are accessible for a given QL. Colored lines (blue to red) represent the QL-dependent boundary below which filter shapes are forbidden because of unstable dynamics. We note that νωL, ν∞ → 0 = QL. An intrinsic frequency exists in region above the QL = 1/2 boundary. A voltage resonance exists in the region above the QL = 1 boundary. We show the accessible subset of corresponding filter shapes at representative positions within the regions (located at  and ) and at the border between regions (located at νωL, ν∞ = 10−1.5, 100, 101.5). (f) Same type of plot as (c), but for the phase response. π/2 and −π/2 are shown as top and bottom bounding dashed lines for the set of phase responses at each location. The gain and phase for the position denoted by the circle are shown in (a) and (c), and for the star in (b) and (e), respectively.
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Related In: Results  -  Collection

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

pcbi.1004636.g009: The 6 distinct filter shapes in (νωL, ν∞)-space.(c, f) Region 1–6 denote the regions exhibiting qualitatively similar filter shapes. E.g. spiking resonance is by definition region 1 and 2. Not all of these six regions are accessible for a given QL. Colored lines (blue to red) represent the QL-dependent boundary below which filter shapes are forbidden because of unstable dynamics. We note that νωL, ν∞ → 0 = QL. An intrinsic frequency exists in region above the QL = 1/2 boundary. A voltage resonance exists in the region above the QL = 1 boundary. We show the accessible subset of corresponding filter shapes at representative positions within the regions (located at and ) and at the border between regions (located at νωL, ν∞ = 10−1.5, 100, 101.5). (f) Same type of plot as (c), but for the phase response. π/2 and −π/2 are shown as top and bottom bounding dashed lines for the set of phase responses at each location. The gain and phase for the position denoted by the circle are shown in (a) and (c), and for the star in (b) and (e), respectively.
Mentions: For this general case, we can introduce the relative quality factor for the full filter, νωL: = /ν1(ωL)//νlow. The response then depends on the five shape features, νlow, νhigh, ωL, QL, and νωL. Denoting ξ = τw/τc, so that and , we can re-express the response function asν1(ω)νlow=1+iν∞ξωωL1+iξν∞ωωL1-ω2/ωL2+iω/QLωL(46)with dynamic gainνm,1(ω)νlow=1+ν∞ξω2ωL21+ξν∞ω2ωL21-ω2ωL22+ω2QL2ωL2.(47)When ω = ωL, , which implicitly defines ξ in terms of νωL, QL and νhigh and closes the representation. Indeed, with time in units of and gain values relative to νlow, the shape of the filter depends only on this triplet: each of the six regions in (ν∞, νωL)-space defined by the boundaries ν∞ = 1, νωL = 1, and ν∞ = νωL provides filters of a qualitatively similar class (see Fig 9).

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