^{*}

Conceived and designed the experiments: PK FC. Performed the experiments: PK FC. Wrote the paper: PK FC.

The authors have declared that no competing interests exist.

The mean input and variance of the total synaptic input to a neuron can vary independently, suggesting two distinct information channels. Here we examine the impact of rapidly varying signals, delivered via these two information conduits, on the temporal dynamics of neuronal firing rate responses. We examine the responses of model neurons to step functions in either the mean or the variance of the input current. Our results show that the temporal dynamics governing response onset depends on the choice of model. Specifically, the existence of a hard threshold introduces an instantaneous component into the response onset of a leaky-integrate-and-fire model that is not present in other models studied here. Other response features, for example a decaying oscillatory approach to a new steady-state firing rate, appear to be more universal among neuronal models. The decay time constant of this approach is a power-law function of noise magnitude over a wide range of input parameters. Understanding how specific model properties underlie these response features is important for understanding how neurons will respond to rapidly varying signals, as the temporal dynamics of the response onset and response decay to new steady-state determine what range of signal frequencies a population of neurons can respond to and faithfully encode.

Cortical neurons continuously receive input from a large number of excitatory and inhibitory synapses

The net synaptic current to a neuron, obtained from the difference between excitatory and inhibitory components, may be quite small compared to the total level of synaptic input (the sum of these two components) if the majority of excitation is cancelled by inhibition. In this case, although the mean input current may be quite small, the variability introduced into the neuronal responses can nevertheless be large. By changing excitation and inhibition independently, the mean and variance (referred to here as “noise”) of the synaptic input current can be varied independently of each other. It should be noted that although we refer to the variance of input current as “noise”, we do not mean to imply that this signal has no useful function. In fact, one purpose of this study is to further explore the consequences of using noise, or input current variance, as a possible information conduit to the neuron.

Although the presence of noise can limit the information transmission capacity of a neuron or a neuronal population

The effects of noise on firing rates of different integrate-and-fire model neurons have been studied extensively (for examples, see

More recently, the possibility has been raised that noise itself may represent a separate conduit of information in addition to the mean input current to a neuron

In this study we examine the temporal dynamics of neuronal responses to sudden changes in either the mean or variance (noise) of the input current. For this study, we divide the firing response into two stages, the “response onset”, essentially a measure of how quickly the model neuron's firing rate reacts to a change in input, and the “decaying response”, a measure of how quickly the firing rate stabilizes to its new steady-state after a sudden change (this division is introduced mainly for clarity of presentation, as there is no true absolute division between these two stages). We find that the temporal dynamics of the response onset may be predicted based on the underlying membrane potential distribution. For this analysis, we focus primarily on integrate-and-fire models to take advantage of their mathematical tractability, but we also examine a more biologically-realistic conductance-based model. The response onset dynamics of each model differ depending on the choice of model as well as the noise parameters (these findings are in agreement with previous findings

In this study, we primarily examined integrate-and-fire neurons, a type of single-compartment neuron model. The dynamics of the membrane potential, V(t), is governed by_{m} is the total membrane capacitance, and Ψ(V) is a model-dependent function of membrane potential (discussed later in this section). In this study we focus on three well-known integrate-and-fire models, the leaky integrate-and-fire (LIF) model, the quadratic integrate-and-fire (QIF) model

I(t) is the sum of two components, an external component, I_{ext}, analogous to an external current injected through a recording electrode, and a synaptic component, I_{syn}, designed to approximate current arising from _{syn} is the sum of a Gaussian white noise process with variance, σ, and a mean current, I_{m}, filtered through a linear filter with time constant τ_{s},_{s} is the synaptic time constant. In our study, the synaptic time constant varied from 0 to 20 ms, as noted. I_{m} and σ were adjusted so that when comparing behavior of different models, the mean firing rate and decay time (the time it takes for the firing rate to reach steady-state after a change in input, see _{ext} comprises the input signal, either a step in mean or variance, and does not pass through the synaptic filter (see

Due to existence of the noise component in the input current, the time-dependence of an individual neuron's membrane potential is not deterministic. As a result, the membrane potential is described by a probability distribution, P(V,I,t)ΔVΔI, that describes the probability of finding the membrane potential in a range of [V,V+ΔV] when input current is in a range [I,I+ΔI] at time

The FP equation connects any inhomogeneity of the probability flow, J(V,I,t), in configuration space to the change in the local probability distribution over time:_{V}(V,I,t) and J_{I}(V,I,t) are different components of the probability flow vector. The boundary conditions imposed on the Fokker-Planck equation, as well as the Ψ(V) term, are model-dependent. For each model, the mean firing rate is equal to the total probability flow across the spike-threshold (defined by V = V_{th}).

In the LIF model, Ψ_{LIF}(V) is a linear function of membrane potential,_{L} is the membrane conductance of the model. The resting membrane potential, V_{rest} = −74 mV, sets V in the absence of any input current. If V depolarizes above a threshold potential, V_{th} = −54 mV, a spike is instantaneously generated and the membrane potential is set to the reset potential, V_{reset} = −80 mV. For large I_{m}, the LIF firing rate asymptotically approaches a linear dependence on input current. In some situations, the firing rate of the LIF model can be calculated analytically

As its name indicates, Ψ_{QIF}(V) depends quadratically on the membrane potential in the QIF model:_{T} is the minimum current required to fire the neuron. Δ = V_{th}−V_{reset}, determines the onset of spike generation and is inversely proportional to the curvature of Ψ_{QIF}(V) at its minimum, V_{0} _{0} = V_{rest} to match the peak of the membrane potential probability distribution (in the subthreshold regime) to that of the LIF model. The rate of membrane potential change increases with the square of its distance from the resting potential. An action potential occurs when the membrane potential diverges to positive infinity (the dynamics of the model allow this to occur in a finite time interval), after which the membrane potential is reset to negative infinity (although see General Notes on Simulating IF Models). Other parameters were adjusted to make the steady-state firing rate curve of the QIF model similar to the LIF model. The minimum current required to drive the model to fire, I_{T} = g_{L}(V_{th}−V_{rest}), was chosen to match the threshold current of the LIF model.

In the absence of noise (and provided that there is sufficient input current to drive the neuron), the firing rate of the model varies as the square root of the mean input current. This firing behavior matches the observed near-threshold behavior of all type I neurons. The QIF model can be mapped to the much-studied θ-models

For the EIF model, first proposed by Fourcaud-Trocmé et al. _{EIF}(V) consists of a linear and an exponential term,_{th}−V_{reset}) is important for determining action potential onset. Its value was chosen to match the asymptotic steady-state firing rates (for large input current) of the LIF model. In the large V limit, Ψ_{EIF}(V) grows superlinearly, causing V to diverge to positive infinity after sufficient depolarization. Also as the QIF model, the divergence of the membrane potential represents an action potential, but V is reset to V_{reset} after an action potential. We set V_{T} = 2V_{th}−V_{reset} to match the threshold input current (in the absence of noise) to that of the LIF model. Also, V_{0} = 2V_{rest} so that in the absence of any additional current or noise, the subthreshold behavior of the EIF model is similar to the LIF model.

In contrast to the QIF model, the firing rate of the EIF is approximately linear for large input (the precise dependence is I_{m}/log(I_{m})).

Model neurons were simulated using a fourth-order Runge-Kutta method. For the purposes of this study, firing rate was measured as the population firing rate of 10^{5} to 10^{6} identical neurons. I_{ext} and I_{m} were identical for each neuron in the population, but the noise component was random and different for each neuron.

When possible, we matched the parameters of the integrate-and-fire models. The membrane time constant, τ_{m} = 20 ms, and the membrane conductance, g_{L}, are equal across all integrate-and-fire models. V_{rest} (for the LIF model) and V_{0} (for QIF and EIF models) were set equal to each other so that the locations of membrane-potential probability distribution peaks (in the sub-threshold regime) for different integrate-and-fire models were matched. As already noted, all other parameters were chosen to make the firing rate curves of the model as similar as possible. As a result, the models require identical threshold current for spiking, and the asymptotic dependence of firing rate on constant input current is the same for the EIF and LIF models (up to a logarithmic factor).

The membrane potential divergence (spiking mechanism) for the QIF and EIF models cannot be achieved numerically because it involves infinitely large potentials. Instead, we defined a large upper bound potential for the EIF model and a large upper and lower bound for the QIF model. The dynamics outside these boundaries, where the effect of noise is negligible, was replaced using approximate analytical expressions

For a more biologically-realistic model, we also studied a conductance-based model proposed by Connor et al. _{syn} and I_{ext}:_{L}, E_{K}, E_{Na}, E_{A} and g_{L}, g_{K}, g_{Na}, g_{A} are the reversal potentials and maximal conductances of a membrane leak conductance, a delayed-rectifier potassium conductance, a fast transient sodium conductance, and a transient A-type potassium conductance, respectively. The dynamics of these conductances is described by five gating variables: n, m, h, a, and b. These gating variables, x^{j} = (n, m, h, a, b), all satisfy a simple first-order differential equation

We seek to examine the temporal dynamics of firing responses to input signals embedded in either the mean or the variance (referred to here as “noise”) of the input current. Common methods of quantitatively studying signal transmission include examining the firing-rate response of neurons to step functions in their inputs _{ext}, and studying the firing responses of neuron models to steps in the mean and in the variance (noise) of I_{ext}. Like I_{syn}, I_{ext} is the sum of a mean current and a Gaussian white noise (see _{ext} does not pass through the synaptic filter and thus is unaffected by τ_{s} (the synaptic time constant – see

We examine two basic features of IF model responses to steps in input signals: the “response onset” and subsequent “decaying response”. Understanding what factors modulate the “response onset” provides insight into how quickly the firing rate of a neuron or a population of neurons can react to time-varying input. Any components in the input that vary faster than the time scale of the response onset will be suppressed in the neuronal firing response. The “decaying response”, on the other hand, describes the approach of the neuronal firing rate to a new steady-state value. This response component is a measure of how quickly a network “forgets” a change in input signal. Any signal that varies at time scales slower than the population response decay time will be reflected faithfully in the population firing rate.

Because Ψ_{LIF}(V) is linear for the leaky integrate-and-fire (LIF) model (see _{I}(V,I,t). The membrane conductance of the model is referred to as g_{L} and the membrane time constant, τ_{m}. I is the mean input current to the neuron, I_{m}+〈I_{ext}〉. The synaptic input current consists of I_{m}, the mean, and a Gaussian white noise process with variance σ^{2}, filtered through a linear filter with time constant τ_{s}. The LIF model mean firing rate, ν(t), in this dimensionally reduced form, is equal to boundary value J at spike threshold, J(V_{th},t). For the LIF model, the probability flow can be written as

The top two panels of _{s} of 0 ms (_{s} is 5 ms, P(V_{th}) is greater than zero (_{th}) is a monotonically increasing function of τ_{s} that vanishes in the limit of τ_{s}→0 (also see _{th},t), in the drift term _{I}ν,_{th}) varies as a function of τ_{s} and mean input current for low (_{th}) correspond to case where I_{m} is just below the value required to fire the neuron and reflect the hypersensitivity of the firing rate responses at this point to any changes in input. As would be expected, this peak becomes less pronounced and the firing response less sensitive to input parameters as the noise magnitude increases (compare

A and B) Membrane potential probability distributions with (A) τ_{s} = 0 ms or (B) τ_{s} = 5 ms. I_{m} was adjusted to that the overall firing rate was 20 Hz. The variance (σ) of the noise was 640 mV^{2}-ms. In (A), the nonzero value of P(V) at V = V_{th} arises from the finite time steps that we use by necessity in our simulations. C and D) The value of the probability distribution at spike threshold, P(V_{th}), as a function of I_{m} and τ_{s} under (C) low noise and (D) high noise conditions. E and F) Absolute value of the first derivative of the probability distribution at threshold, |∂P(V_{th})|. In the low noise regime the variance of the synaptic component was 160 mV^{2}-ms and in the high noise regime it was 1440 mV^{2}-ms. (Input is given in mV, the resulting membrane potential depolarization).

The trends demonstrated in this figure suggest that a comparably bigger instantaneous response to a mean current jump will be evoked for larger values of τ_{s}. In the top two panels of _{s} = 0 ms (_{s} = 5 ms (

Each panel is the firing rate of an LIF neuron in response to A) a step in mean input current with τ_{s} = 0 ms, B) a step in mean input current with τ_{s} = 5 ms, C) a step in input current noise with τ_{s} = 0 ms, or D) a step in input current noise with τ_{s} = 5 ms. In (A) there exists a small instantaneous jump that arises because of the finite time steps used in our simulations. For panels (A–C), the variance of the synaptic component was 1440 mV^{2}-ms (prior to the input step). In (D), the variance of the synaptic component was 1000 mV^{2}-ms and the variance of the external input (prior to the step) was 40 mV^{2}-ms.

A jump in mean current pushes the peak of the probability distribution towards spike threshold, instantaneously increasing the probability flow at threshold and inducing an instantaneous jump in firing rate. However, when τ_{s} is very small (for example see _{s} = 0 ms), the total firing rate is dominated by the diffusive part of the probability flow (both the diffusive and the drift parts of the probability flow depend on I_{m}, see Text S1), and the resulting instantaneous jump in firing rate, δ_{I}ν, is negligible compared to the final change in firing rate after the probability distribution reaches its new steady state. Because of the dominance of the diffusive component of probability flow, the firing rate of the LIF model to a small jump in mean current approaches its final steady-state from below (see _{m},σ), is primarily determined by the mean input current and only weakly by noise magnitude.

A small jump in noise amplitude, δσ, also results in an instantaneous jump in firing rate, δ_{σ}ν,_{th}) as it appears in the diffusion term of the probability flow (see eq. 11). The behavior of −∂P(V_{th}) determines the response to a step in noise _{th}) varies as a function of τ_{s} and I_{m}. Comparison of _{th}) on τ_{s} is more complex than for P(V_{th}). As before, the peaks in the plots correspond to the condition in which the neuron is just below firing threshold and extremely sensitive to changes in input. In the sub-threshold regime, increasing τ_{s} causes an increase in the magnitude of −∂P(V_{th}). In the superthreshold regime, however, there is a range in which −∂P(V_{th}) decreases with increases in τ_{s}. This range corresponds to the situation in which the mean current is far above threshold. Because this trend only occurs for a very small set of parameters that do not correspond to a biologically-realistic situation, we did not investigate it further.

_{V}P is coupled to the magnitude of the noise, σ, the response onset to a jump in noise is always associated with an instantaneous increase in firing rate that “overshoots” the final steady-state value. Examples of this “overshoot” behavior can be seen in _{s} always causes an increase in the magnitude of −∂P(V_{th}), enhancing the magnitude of the overshoot. The increase in noise magnitude eventually acts to flatten the probability distribution, decreasing the absolute value of −∂_{V}P at firing threshold. The net increase in steady-state firing may thus be relatively small.

Our results demonstrate that sudden small changes in input current will evoke different firing rate changes, δ_{I}ν and δ_{σ}ν, depending on whether the change is in the mean, δI, or the variance, δσ. Although in this study we focus on using equations (12) and (13) to connect jumps in input to the firing rate behavior during response onset, these equations hold for other patterns of time-varying input.

Whereas the firing rate of the LIF model is equal to the probability flow at firing threshold, the mean firing rate of the QIF model is equal to the probability flow at infinity. The probability distribution of the QIF model is shown in

A) Probability distribution of the QIF model membrane potential. B) Probability distribution of the EIF model. For both panels, τ_{s} = 0 ms and the variance of synaptic component was 9000 mV^{2}-ms, resulting in an average firing rate of 20 Hz.

A) Response to a step of mean input current with τ_{s} = 0 ms. B) Response to a current step with τ_{s} = 5 ms. C) Response to a step in noise for τ_{s} = 0 ms. D) Response to a step in noise for τ_{s} = 5 ms. For (A) and (B), the variance of synaptic component was 36000 mV^{2}-ms. For (C) and (D), the variance of synaptic component (prior to the noise step) was 4000 mV^{2}-ms.

Similar to the QIF model, the response onset of the EIF model also does not contain an instantaneous component. The probability distribution of the EIF model is given in

A) Response to a current step with τ_{s} = 0 ms. B) Response to a current step with τ_{s} = 5 ms. C) Response to a step in noise for τ_{s} = 0 ms. D) Response to a step in noise for τ_{s} = 5 ms. As in ^{2}-ms. For (C) and (D), the variance of the synaptic component (prior to the noise step) was 4000 mV^{2}-ms.

Previous work and the results discussed in the previous sections show that the action potential threshold mechanism appears to play a critical role in the response onset

The probability distribution of the conductance-based model, while firing, is plotted in this reduced representation in

A) Probability distribution of the conductance-based model, plotted against membrane potential (V) and the potassium gating variable (n). The variance of the synaptic component was 1000 mV^{2}-ms. B) Firing rate of the conductance-based model in response to a step of input current. The synaptic time constant, τ_{s}, was 0 ms. The variance of synaptic component was 4000 mV^{2}-ms. C) Firing rate of the conductance-based model in response to a step of noise, with τ_{s} = 0 ms. The variance of synaptic component (prior to the noise step) was 2250 mV^{2}-ms.

Because the spike-generation mechanism of the conductance-based model is very fast relative to the temporal dynamics of the subthreshold membrane potential, only a small subpopulation of neurons exists in the action potential regime at any time, including the regime near spike-detection threshold. As with the QIF and EIF models, the response onset following a step in mean or noise input does not have an instantaneous component (see

A general feature of firing responses to step functions in either mean or variance displayed by all models in this study is a decaying oscillation towards the new firing rate value. Any jump in input creates a disparity between the probability distribution profile (the steady-state solution immediately before the jump) and the new steady-state solution. For a population of neurons, this initial imbalance has a synchronizing effect and creates oscillations in the firing rate across the population

The noisy component of the input current eventually cancels the mismatch between the steady-state probability profiles before and after the input step by allowing the potential distribution to asymptotically approach the new steady-state distribution. The higher the magnitude of the noise, the faster the firing rate relaxes to its final steady firing rate. For relatively small jumps in input current parameters, it is possible to asymptotically fit the firing rate with only one decaying component,_{decay} describes the time scale of relaxation. The thin black lines in

For the top panels, the jumps in firing rate were driven by steps in mean input current. For the bottom panels, the model neurons are responding to steps in noise. In panels (A) and (C), τ_{s} = 0 ms and the variance of synaptic component was 10 mV^{2}-ms. In panels (B) and (D), τ_{s} = 5 ms. Prior to the step in noise, the variance of synaptic component was 90 mV^{2}-ms in (C) and 40 mV^{2}-ms for the variance in synaptic component and 10 mV^{2}-ms for the external input variance in (D).

The top panels are QIF firing rates in response to jumps in mean input current and the bottom panels are QIF firing rates in response to jumps in noise. For panels (A) and (C), τ_{s} = 0 ms. For panels (B) and (D), τ_{s} = 5 ms. The variance in synaptic component was 4000 mV^{2}-ms for (A) and (B), or 2250 mV^{2}-ms prior to the step in noise for (C) and (D).

The top panels of EIF firing rates in response to jumps in mean input current and the bottom panels are EIF firing rates in response to jumps in noise. As in _{s} = 0 ms, and for panels (B) and (D), τ_{s} = 5 ms. As in ^{2}-ms for (A) and (B), or 2250 mV^{2}-ms prior to the step in noise for (C) and (D).

The firing-rate dynamics of our models can be understood by studying the Fokker-Planck equation that governs the dynamics of the probability distribution, P(V,t). The Fokker-Planck operator L_{FP} explicitly depends on the input-current mean and variance. The spectrum of the FP operator, λ_{0}(t), λ_{1}(t), … , defines a hierarchy of time scales. For time scales Δt that are much larger than 1/|Re(λ_{2}(t))| the dynamics of FP equation can be replaced by a simple oscillator. In particular, the firing rate of our noisy population is the real part of ν(t) in the following first-order differential equation_{1}(t) while the decay time constant is related to the inverse of the real part of λ_{1}(t).

Interestingly, the relationship between τ_{decay} and noise magnitude follows a power law for a large range of parameters_{decay} and noise magnitude can be understood through a perturbative calculation of the first non-zero eigenvalue of the Fokker-Planck equation for small magnitudes of noise. We can break the Fokker-Planck operator L_{FP}(I,σ) into a noise-independent component and a noise-dependent component, i.e. L = L_{0}+σ^{2}L_{1}. The appearance of the multiplicative σ^{2} term causes the perturbative expansion of all eigenvalues in increasing powers of σ^{2}. In particular, the real part of the first non-zero eigenvalue is dominated by a σ^{2} term in the small noise limit._{decay} = 1/|Re(λ_{2})| that was introduced in the above equation can be used to explain the power-law dependence of τ_{decay} on σ. The analysis for the QIF model is drastically simplified because the whole parameter space (I,σ) can be mapped by scaling time and membrane potential to three 1-dimensional subspaces (I = −1,0,+1,σ)

LIF τ_{decay} (A), QIF τ_{decay} (B), and EIF τ_{decay} (C) are given as functions of final noise magnitude (noise level after the jump in noise). For the QIF model (B), the decay time constants measured from responses to a jump in mean are given by empty squares and the decay time constant measured from responses to jumps in noise are given by filled circles. τ_{0} = 1 ms and σ_{0}^{2} = 0.1 mV^{2}-ms.

When the jump in input is large relative to the pre-jump value, the initial response overshoots the expected decaying oscillation for both QIF and EIF models (for examples, see

Because the quadratic term in the Ψ(V) function dominates spike generation in the QIF and EIF models, differentiating between their firing responses can be difficult. For each model, we adjusted Δ to set the Ψ(V) functions of the QIF and EIF models to have the same radius of curvature at their minimum (see

We also studied the responses of the conductance-based neuron to sudden jumps in mean and noise. The initial response to a jump in either mean or noise begins with a sharp onset (discussed earlier) followed by a decaying oscillation, as shown in

A) Firing rate of the conductance-based model in response to a step of input current. The synaptic time constant, τ_{s}, was 0 ms. The variance of synaptic component was 722.5 mV^{2}-ms. B) Firing rate of the conductance-based model in response to a step of noise, with τ_{s} = 0 ms. The variance of synaptic component was 160 mV^{2}-ms.

We have studied the temporal dynamics of the firing rate response of integrate-and-fire and conductance-based models to rapid changes in mean or noise. For analysis purposes, we divided the time course of the population response into two regimes. The initial response, “response onset”, indicates how fast the population reacts to a change in its input. The asymptotic behavior of the response as it approaches its final steady-state value, referred to in this paper as the “decaying response”, is described by a characteristic time scale, τ_{decay}. Any signals with time scales slower than τ_{decay} will be reflected in population firing rate with little distortion.

The temporal firing rate response of an integrate-and-fire model can be predicted based on the characteristics of the membrane potential probability distribution near threshold and the coupling between the probability flow and the input current (for a review see _{s}. Because this instantaneous component arises from a non-zero value of the probability distribution at spike threshold, it is absent when τ_{s} equals zero. For a jump in noise, the LIF response onset always contains an instantaneous component and overshoots the final steady-state firing rate. Within the range of firing rates that we studied, the size of the response onset increases for larger synaptic time constants due to increases in the values of both the probability distribution of the membrane potential and its derivative at spike threshold for larger values of the synaptic time constant.

The firing rates of the QIF, EIF, and conductance-based models, on the other hand, change smoothly, even in response to an instantaneous increase in input current. This property is due to the fast decay of the membrane potential distribution at relatively depolarized (and thus close to spike detection threshold) values. Silberberg et al have previously shown that living neurons also respond to a step in noise with a rapid rise in firing rate

All IF model responses to relatively small jumps in mean current or noise in the asymptotic region can be fit to exponentially decaying oscillations for small τ_{s} (i.e. τ_{s}≪τ_{m}). The decay time constant has a power law dependence on the magnitude of the background noise. We focused on the firing response of various IF neurons at t = 0 (response onset), and at t→∞ (decaying response). For the parameter range we studied, QIF and EIF responses to large steps in mean or noise cannot be fit to a simple decaying oscillation due to the importance of more rapidly decaying modes. For small input jumps, the fit matches quite well, although there are overshoots near t = 0. Also, the responses tend to decay faster and appear sharper after a jump in noise than a jump in mean. This sharpening of the response is due to the increased level of noise. As mentioned earlier, overshoots arise through the contributions of higher harmonics (eigenfunctions). The expansion coefficients of these rapidly decaying modes (a_{n} in equation 9 of Text S1) decrease with a power law as a function of n for large n, i.e. lim_{n→∞} a_{n}∝ n^{−β}. This relation is due to the existence of ∂^{2}/∂V^{2} (the curvature of a function) in the Fokker-Planck operator, L_{FP}, which makes higher eigenfunctions more oscillatory functions of V. The summation of these faster modes adds up to the sharper appearance of oscillation just after the jump.

An increase in noise reduces the decay time constant, allowing the firing rate to more faithfully follow the input current. This process is much like “dithering”, a technique used to minimize artifacts in signal transmission. We can define the error in transmission of a jump in mean or noise as the average in a time window T of the difference between 1 and relative final firing rate ν(t)/ν(t_{∞}). This parameter was named the “dissimilarity” parameter for the more general case of an arbitrary input _{opt}, that optimizes signal transmission.

The optimal value of noise will depend in part on the time scale of the encoded signal. Any input signal can be approximated by a piece-wise constant function with jumping period of T. The variables σ_{opt} and T are dependent since T appears in a factor of 1-exp(T/τ_{decay}) in the dissimilarity parameter if eq. (10) approximates the firing rate well at all times. The weak dependence of σ_{opt} on T in the large noise limit can be the basis for a robust mechanism of fast and faithful signal transmission. In contrast, in the small noise regime T and σ_{opt} are strongly correlated and optimizing signal transmission requires that the system adjust the magnitude of noise.

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