Neural Noise Can Explain Expansive, Power-Law Nonlinearities in Neural Response Functions

Abstract

Many phenomenological models of the responses of simple cells in primary visual cortex have concluded that a cell's firing rate should be given by its input raised to a power greater than one. This is known as an expansive power-law nonlinearity. However, intracellular recordings have shown that a different nonlinearity, a linear-threshold function, appears to give a good prediction of firing rate from a cell's low-pass-filtered voltage response. Using a model based on a linear-threshold function, Anderson et al. showed that voltage noise was critical to converting voltage responses with contrast-invariant orientation tuning into spiking responses with contrast-invariant tuning. We present two separate results clarifying the connection between noise-smoothed linear-threshold functions and power-law nonlinearities. First, we prove analytically that a power-law nonlinearity is the only input-output function that converts contrast-invariant input tuning into contrast-invariant spike tuning. Second, we examine simulations of a simple model that assumes instantaneous spike rate is given by a linear-threshold function of voltage and voltage responses include significant noise. We show that the resulting average spike rate is well described by an expansive power law of the average voltage (averaged over multiple trials), provided that average voltage remains less than about 1.5 SDs of the noise above threshold. Finally, we use this model to show that the noise levels recorded by Anderson et al. are consistent with the degree to which the orientation tuning of spiking responses is more sharply tuned relative to the orientation tuning of voltage responses. Thus neuronal noise can robustly generate power-law input-output functions of the form frequently postulated for simple cells.

Artificial Synaptic Conductances Reduce Subthreshold Oscillations and Periodic Firing in Stellate Cells of the Entorhinal Cortex

“…To account for this difference it is important to consider that preventing the cell from firing under the synaptic-mediated fluctuations requires significant hyperpolarization compared with control. This is because of the lowering of spike rheobase (input current required to elicit spike firing) and smoothing of the frequency-current relationship under noisy conditions (Barbi et al, 2000;Longtin, 2000;Chance et al, 2002;Miller and Troyer, 2002). The membrane fluctuations associated with background synaptic activity permit occasional threshold crossings despite maintaining a mean membrane voltage that is significantly more negative than that required to elicit spikes in the absence of noisy synaptic stimuli.…”

“…This results from the fact that under noisy conditions the cell must be held at a more negative mean membrane voltage to maintain the same firing frequency as the control condition (Barbi et al, 2000;Longtin, 2000;Chance et al, 2002;Miller and Troyer, 2002). The fluctuating nature of the synaptic stimulus protocol permits occasional excursions past spike threshold that would not be possible if the cells were held at the same mean voltage under quiescent conditions.…”

The contribution of spike threshold to the dichotomy of cortical simple and complex cells

“…Here, we have modeled the membrane response of each neuron as the sum of a steady elevation (V 0 ) and a sinusoidal modulation (V 1 ); this response is transformed to firing rate by a power law (Fig. 1a) in which firing rate is proportional to the voltage above rest raised to the power of 2 (p = 2) 14,15 .…”

“…Most neurons, including neurons of primary visual cortex, do behave in a thresholdlinear fashion under controlled conditions such as the repeated injection of current pulses 31 . The large trial-to-trial variability in the responses of neurons of primary visual cortex, however, tends to smooth the average relationship between mean membrane potential and mean firing rate so that it approximates a power law 14,15,32,33 . Additional smoothing might originate from variations in threshold related to dV/dt 34,35 .…”

Orientation selectivity in visual cortex by fluctuation-controlled criticality

“…Much work has focused on the smoothing effect of fluctuations on the transfer of intracellular currents to extracellular spiking, and their relation to contrast invariant orientation tuning (14,28,29). Although our fluctuation-dominated network model of V1 shares these properties, its operating point is one of near-criticality, at the onset of a bifurcation of multistability and hysteresis, itself controlled by the level of intrinsic fluctuations in the network.…”