The qualitative shape of voltage response depends on <i>Q</i>_<i>L</i>.

<p>Here we classify the current-to-voltage filter shapes shown as colored solid lines in (a), (b), and (c), which show the three <i>Q</i>_<i>L</i>-regimes with respective examples for <i>Q</i>_<i>L</i> = 0.1, 0.75, 10. In each plot, the high pass component of the voltage response is shown as the colored dashed lines, one for each of three representative values of its characteristic frequency, <i>ω</i>_<i>L</i>τ_<i>w</i> = 10² > γ(blue), <i>ω</i>_<i>L</i>τ_<i>w</i> = 1(green), and <i>ω</i>_<i>L</i>τ_<i>w</i> = 10⁻² < <i>γ</i>⁻¹(red). The solid black line is the low pass component of the voltage response. For the regime shown in (a), the green case can not be achieved when <i>w</i> is hyperpolarizing (<i>g</i> > 0) and the example red case cannot be achieved because it violates the stability condition <i>Q</i>_<i>L</i> < <i>ω</i>_<i>L</i>τ_<i>w</i>.</p

The accessible region of filter shapes depends on <i>Q</i>_<i>L</i> and the relative speed of spiking to intrinsic dynamics <i>ξ</i> = τ_<i>w</i>/τ_<i>c</i>.

<p>The purple region marks the region of voltage resonant filters. This region is contained in the red region of stable filters, whose lower bound moves to larger <i>ν</i>_{<i>ω</i>_<i>L</i>} with <i>Q</i>_<i>L</i>. For relatively slow intrinsic spiking (a, b, c), there are regions of non-spiking resonant(<i>ν</i>_∞ > <i>ν</i>_{<i>ω</i>_<i>L</i>}), but voltage resonant filters. Filters for relatively fast intrinsic dynamics (d, e, f) only exist as high pass resonant filters for large <i>Q</i>_<i>L</i>. (Left to right: , 1.1. Top row: <i>ξ</i> = 10. Bottom row: <i>ξ</i> = 0.1).</p

Correspondence of response between analytical result of no-reset model (blue line) and the numerical result of its EIF version (black circles).

<p>The correspondence holds up to a high frequency cut-off, <i>f</i>_<i>limit</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004636#pcbi.1004636.e207" target="_blank">Eq (54)</a>), due to finite rise time of action potential controlled by Δ_<i>T</i> = 0.35, 0.035. The EIF-version was simulated with <i>V</i>_<i>thr</i> = 1.15, 3, and <i>V</i>_<i>T</i> = 0.8, −1 (the latter was adjusted to keep <i>ν</i>₀ = 2Hz fixed). The black dashed lines correspond to the high frequency limit of the response of the EIF-type model (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004636#pcbi.1004636.e206" target="_blank">Eq (53)</a>). The no reset model had the default parameters.</p

Complete Firing-Rate Response of Neurons with Complex Intrinsic Dynamics

<div><p>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.</p></div

Regimes of the current-to-voltage transfer function.

<p>(a) Phase diagram of the transfer function. The region of depolarizing <i>w</i> (low frequency amplifying, <i>V</i>_<i>low</i> > 1) is shown in purple and voltage resonance in green. The filter is unstable in the blue region. An intrinsic frequency exists above the dotted line, <i>Q</i>_<i>L</i> = 1/2. Note that there is a region with <i>Q</i>_<i>L</i> > 1/2 and no voltage resonance, and vice versa. The star and circle denote the example values of (<i>ω</i>_<i>L</i>τ_<i>w</i>, <i>Q</i>_<i>L</i>) used in (b) and (c), respectively. (b) An example of the current-to-voltage filter in the case of resonance with no intrinsic frequency (τ_<i>V</i> = 10, τ_<i>w</i> = 100, <i>g</i> = 1.2). (c) An example of the current-to-voltage filter in the case of no voltage resonance despite the existence of an intrinsic frequency (τ_<i>V</i> = 10, τ_<i>w</i> = 5, <i>g</i> = 0.5). The rising and falling dashed lines in (b) and (c) denote the contributions of the high pass, , and the low pass, , respectively. Their combination forms the current-to-voltage filter, which are shown as solid lines.</p

An example of filter shaping: attenuation at high frequencies uncovers an amplified band of intermediate frequencies.

<p>(a) The shape space representation showing the region of accessible filters (white) for <i>Q</i>_<i>L</i> = 0.1. The blue regions exhibit unstable filters. Filters obtained from points above the thick black line are spiking resonant. Filters obtained from points above the black dashed line are voltage resonant. The arrow illustrates a path in shape space along which <i>ν</i>_∞ is decreased. (b) and (c) show the beginning and end filters along the path in (a). For (b) and (c), blue dashed lines are the high and low pass components of the current-to-voltage filter, which itself is shown in solid blue. Shown in red is the voltage-to-spiking filter which combined with the current-to-voltage filter gives the full filter, shown in black.</p

Effect of model parameters on the fluctuation-driven stationary response.

<p>The stationary firing rate, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004636#pcbi.1004636.e057" target="_blank">Eq (18)</a> for <i>I</i>₀ ∼ 0 (a) increases monotonically with the strength of input fluctuations and (d) decreases monotonically with the intrinsic frequency. Across each of τ_<i>I</i> and τ_<i>w</i> ((b) and (c) respectively), the rate exhibits a maximum. Insets are the mean input dependent expression for the stationary response, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004636#pcbi.1004636.e053" target="_blank">Eq (17)</a>, valid in the regime <i>I</i>₀ ≪ 1. Inset color refers to the value of the parameter (<i>σ</i>_<i>I</i>, τ_<i>I</i>, τ_<i>w</i> and Ω) at the location of the colored dots in the main plots. Parameters were otherwise set to their default values.</p

The type of <i>w</i>-current depends on the values of the intrinsic parameters.

<p>(a) Intrinsic parameter phase diagram in (τ_<i>w</i>/τ_<i>V</i>, <i>g</i>). <i>w</i> can be depolarizing (<i>g</i> < 0) or hyperpolarizing (<i>g</i> > 0). <i>w</i> contributes an intrinsic frequency to the model in the colored region. The dynamics are unstable if <i>g</i> < −1. Iso-Ω lines are shown in white ( for large τ_<i>w</i> and for small τ_<i>w</i>). (b) When , the phase diagram can be cast in (τ_<i>w</i>/τ_<i>V</i>, <i>Ωτ</i>_<i>V</i>)-space. Iso-<i>g</i> lines are shown in white. (See [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004636#pcbi.1004636.ref029" target="_blank">29</a>] for a similar plot).</p

The 6 distinct filter shapes in (<i>ν</i>_{<i>ω</i>_<i>L</i>}, <i>ν</i>_∞)-space.

<p>(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 <i>Q</i>_<i>L</i>. Colored lines (blue to red) represent the <i>Q</i>_<i>L</i>-dependent boundary below which filter shapes are forbidden because of unstable dynamics. We note that <i>ν</i>_{<i>ω</i>_<i>L</i>, <i>ν</i>_∞ → 0} = <i>Q</i>_<i>L</i>. An intrinsic frequency exists in region above the <i>Q</i>_<i>L</i> = 1/2 boundary. A voltage resonance exists in the region above the <i>Q</i>_<i>L</i> = 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 <i>ν</i>_{<i>ω</i>_<i>L</i>}, <i>ν</i>_∞ = 10^−1.5, 10⁰, 10^1.5). (f) Same type of plot as (c), but for the phase response. <i>π</i>/2 and −<i>π</i>/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.</p

Parameter groups for each dynamics.

<p>Parameter groups for each dynamics.</p