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Wrote the paper: CL BD.

The authors have declared that no competing interests exist.

The modulation of the sensitivity, or gain, of neural responses to input is an important component of neural computation. It has been shown that

Many neural computations, including sensory and motor processing, require neurons to control their sensitivity (often termed ‘gain’) to stimuli. One common form of gain manipulation is divisive gain control, where the neural response to a specific stimulus is simply scaled by a constant. Most previous theoretical and experimental work on divisive gain control have assumed input statistics to be constant in time. However, realistic inputs can be highly time-varying, often with time-varying statistics, and divisive gain control remains to be extended to these cases. A widespread mechanism for divisive gain control for static inputs is through an increase in stimulus independent membrane fluctuations. We address the question of whether this divisive gain control scheme is indeed operative for time-varying inputs. Using simplified spiking neuron models, we employ accurate theoretical methods to estimate the dynamic neural response. We find that gain control via membrane fluctuations does indeed extend to the time-varying regime, and moreover, the degree of divisive scaling does not depend on the timescales of the driving input. This significantly increases the relevance of this form of divisive gain control for neural computations where input statistics change in time, as expected during normal sensory and motor behavior.

Gain modulation (or gain control) is an adjustment of the input-output response of neurons, and is widely observed during neural processing

Noise induced phenomena in nonlinear systems are a rich avenue of study

These studies address the gain control of a transfer function where the signal is either static or statistically stationary and the neural output is the time averaged firing rate. However, many neural coding tasks involve the processing of time-varying, high frequnecy stimuli. In these situations neural response are often transient, and a quasi-static approximation of input-output transfer fails to capture the actual spike response. For example, in the rodent vibrissa sensory

Any theoretical treatment of this problem requires 1) a framework accurately capturing the time varying spike response owing to time varying input statistics (e.g. temporally inhomogeneous input and output statistics), and 2) sufficient biophysical detail to incorporate conductance based synaptic inputs within spike creation. A useful tool for incorporating these two features into neuron models is the population density method

We consider a leaky integrate-and-fire neuron (LIF) driven by a pre-synaptic population of excitatory (e) and inhibitory (i) cells. The neuron's voltage change is given by a random differential equation:

We decompose the pre-synaptic input into a time-inhomogeneous ‘driver’ term

Of interest are the output threshold crossing times, and we estimate response statistics by combining the responses from

^{−1}, 1400 s^{−1}, and 1900 s^{−1} (with corresponding ^{−1}, and 1847.40 s^{−1}, respectively), which we respectively label

(A) Excitatory driving input with rate

(A) The response curve

A Monte Carlo simulation of Equation (2) would be computationally expensive to ensure an accurate result. In many studies only qualitative effects are reported, and thus quantitative accuracy is not at a premium. However, in our study the accuracy demands are large, as we will quantitatively compare the time dependent

In the population density method, neurons with similar biophysical properties are grouped together, and the evolution of a density function

Let

Gain modulation is typically studied in the equilibrium regime

The nonequilbirum response

We compute the equilibrium input/output relationship,

In contrast, for very high output firing rates divisive gain modulation does not occur. The responses

In summary, population density methods (section: Population Density Approach) can replicate fluctuation-induced divisive gain modulation of the equilibrium response at low firing rates, previously observed in: simple integrate-and-fire models

We study the influence of background fluctuations on the nonequlibrium response to a highly time-varying excitatory drive. We choose an input rate

(A) The driving input ^{−1} and

The main result of our study is that fluctuation-induced divisive gain modulation is robust for low to moderate output firing rates in response to dynamic stimuli, despite the complicated dynamics of the leaky integrate-and-fire neuron in the nonequilibrium regime (i.e

Previously, Holt & Koch

(A) The response with the same driver input as before (

When the dynamic stimuli are increased so that resulting output firing rates are larger, the neurons no longer exhibit divisive gain modulation. Increasing the overall intensity of the driving input

(A) Top Panel: Same stimulus in

In our model, when the driving input rate ^{−1}, depending on the background level of activity. Although extracellular recordings in the cortex suggest the neurons can fire spontaneously at rates larger than 2 s^{−1} ^{−1}

(A) The logarithm of the area (or error) between the time-dependent response curve scaled by the equilibrium scale factor ^{−1} marked by stars (*) correspond to the difference in area between the curves in ^{−1} correspond to the difference in area between the curves in

Divisive gain modulation with dynamic stimuli is robust in the subthreshold regime (^{−1} correspond to the difference in area between the curves in ^{−1} correspond to the difference in area between the curves in

The average (unscaled) time-dependent response of the neurons with the same parameters and driver inputs as ^{−1} but the firing rate response can be quite low and as high as 15 s^{−1} (see

A gain control scheme will be effective in unpredictable environments if it is

The red line is the optimal scaling factor

To better describe the mechanism underlying fluctuation-induced divisive gain control in the nonequlibrium, we focus on a weak time modulation of the input drive and compute the linear frequency response

Our earlier results (

(A) Fluctuation driven regime. Top Panel: the frequency response is flat up to very large frequencies for all 3 different levels of balanced background activity. In all 3 cases, there was a constant excitatory

We remark that the near exact scaling of

When the neurons are in the drift dominated regime, the frequency responses does not scale in a multiplicative manner because there are resonant peaks at integer multiples of the steady-state firing rate

Chance et al.

The analysis of the time dependent response for weak signals showed how a scaling of

Divisive gain control is a central tool in many neural computations

The LIF model we have used is an approximation to the dynamic clamp experiments of Chance et al.

The population density equations (6)–(8) that characterize the LIF model contain a partial differential-integral equation that is difficult to analyze. Our model is more general than white noise models that have an advection/diffusion density equation (e.g, Fokker-Planck equation) because it allows for large voltage changes upon receiving synaptic input events. However, the simulations shown in this paper are in the regime where the diffusion approximation is good. If the voltage change upon receiving synaptic events (excitatory or inhibitory) is assumed to be small, a good diffusion approximation of (6)–(8) is obtained by replacing

The advection/diffusion coefficients. (A) The functions d^{e/i}_{0}(v). (B) The functions d^{e/i}_{1}(v). Parameters: PSP = +0.5 or −0.5 mV (see main text for an explanation), τ_{m} = 20 ms, ε_{i} = −80 mV, ε_{r} = −70 mV, V_{th} = −55 mV, ε_{e} = 0 mV.

(8.28 MB TIF)

Fokker-Planck approximation to full density equations

(0.01 MB TEX)

This work was started at the Courant Institute of Mathematical Sciences at NYU (CL & BD), and the Center for Neural Science at NYU (BD). We thank Alex Reyes, Anne-Marie Oswald, and Dan Tranchina for fruitful discussions. BD thanks Jessica Cardin for a motivating discussion.