Computational role of mean-encoded signals.

<p>(Top) Representation of periodic mean stimuli in the population rate of noisy, independent neurons. (Left) Representation of step-like mean signals in the population rate of noisy, independent neurons. (Bottom) Common fluctuating currents from presynaptic partners represent a common mean signal that leads to pairwise spike correlation function . (Right) The average voltage before a spike is shaped by the linear mean response. denotes the input current correlation function. The role of the linear response function is indicated by a dashed green line. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002239#s2" target="_blank">Results</a> obtained in the alternative threshold model are discussed in the indicated Sections of this manuscript.</p

Linear response to mean and variance modulations in a population of independent threshold neurons.

<p>(A) Normalized amplitude vs. in response to mean current modulations, simulations (circles) and analytical results in Eq. 21 (solid line). (B) vs. in response to current variance modulations, simulations (circles) and analytical results in Eq. 24 (solid lines). Regimes of high-pass and low-pass behavior for linear response function for mean (C) and variance modulations (D). Note, vector strength in (A) and (B) is proportional to the linear response , see Eq. 53.</p

Spike generation and signal representation in the single spiking threshold neuron.

<p>(A) Spike generation from a temporally correlated Gaussian voltage trace in a single threshold neuron. (B) Encoding of common signals by the population firing rate of independent threshold neurons. Note, that can be either linearly related to the stimulus (linear regime for weak signals, Eq. 11, 12) or be described by a non-linear response function (e.g. see Eq. 20).</p

Encoding in the mean and variance channel.

<p>(A) Simultaneous increase of excitatory and reduction of inhibitory activity (or vice versa) results in a mean current change (right, green). On the other hand, simultaneous increase (or reduction) in excitatory and inhibitory spiking activity results in modifications of the net current variance (left, red). These modifications constitute two primary channels of communication in a cortical network. (B) In a cortical network the excitatory and inhibitory currents add up such that the net somatic current is only weakly correlated across neurons <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002239#pcbi.1002239-Renart1" target="_blank">[31]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002239#pcbi.1002239-Ecker1" target="_blank">[32]</a>.</p

Demonstration of population firing rate modulation and phase locking.

<p>(A) Simulated population firing rate for mean current modulation for , , and , time bin . This results in , and in the amplitude of the firing rate modulation of . Solid lines denote the envelop of (red) and the current modulation (black). Black and red arrows indicate the phase relation between the input current and the evoked firing rate response. (B) Theoretical distribution of phase lags for varying modulation depth , for illustration we chose (from (A)). The solid curves are the distribution envelop for (red), (black), (blue). The arrows indicate the corresponding mean phase .</p

Symbol nomenclature in the order of appearance.

<p>Symbol nomenclature in the order of appearance.</p

Statistics of spike triggering events in the threshold neurons.

<p>(A) Spike triggered average voltage for and firing rates ; simulated results (circles) and analytical solution in Eq. 37 (solid lines). (B) Spike triggered voltage covariance for , , the cross section is shown at the right. Simulated results (circles) and analytical solution in Eq. 39 (solid lines). The solid vertical black line indicates .</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

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

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