Envelope gating and noise shaping in populations of noisy neurons J. W. Middleton, 1,2,3 E. Harvey-Girard, 2 L. Maler, 2,3 and A. Longtin 1,2,3 1 Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Canada K1N 6N5 2 Department of Cellular and Molecular Medicine, University of Ottawa, 451 Smyth Road, Ottawa, Canada K1H 8M5 3 Centre for Neural Dynamics, 451 Smyth Road, Ottawa, Canada K1H 8M5 Received 31 July 2006; published 28 February 2007 Narrowband signals have fast and slow time scales. The transmission of narrowband signal features on both times cales, by spiking neurons, is demonstrated experimentally and theoretically. The interaction of the narrowband input and the threshold nonlinearity may create out-of-band interference, hindering the transmis- sion of signals in a low-frequency range. The resultant out-of-band signal is the “envelope,” or time-varying modulation of the narrowband signal. The levels of noise and nonlinearity intrinsic to the neuron gate trans- mission on the slow “envelope” time scale. When a narrowband and a distinct slow signal drive the neuron, the slow signal may be poorly transmitted. Increasing intrinsic noise in an averaging network removes the enve- lope in favor of the slow signal, paradoxically increasing the signal-to-noise ratio. These gating effects are generic for threshold and excitable systems. DOI: 10.1103/PhysRevE.75.021918 PACS numbers: 87.19.La, 87.16.Xa, 87.19.Nn, 89.70.c Nonlinear dynamical systems driven by noise and har- monic signals can display a wide range of interesting phe- nomena. This is particularly the case for excitable and threshold systems in, e.g., laser physics and biology 1–4. In physical systems, “harmonic” signals are often more of a narrowband nature, with power over a significant bandwidth see Fig. 1. These have statistical properties intermediate to those of harmonic signals and broadband noise. Narrowband signals have at least two time scales: one related to a fast oscillation, or carrier, and a longer one related to the slow modulation, or envelope, of the carrier. Their effect has been studied in bistable systems 5,6, charge density waves in semiconductors 7 and in coupled Josephson junctions 8. In the field of neuroscience, narrowband signals occur in natural stimuli 9–11; they can, along with other signals, drive large-scale cortical activity 12,13. A recent experi- mental study has further revealed that sensory systems can process the two time scales in parallel 11. Rectification, which linearly transmits only one polarity of an analog signal, is known to be sufficient for extracting an envelope from a narrowband signal in physical systems 14,15. How is this possible in noisy spiking neurons? How are the different time scales transmitted, and how does this interfere with transmission of other slow signals? These is- sues are the focus of this paper. We first demonstrate, using experiments and theory, how the neuron spiking threshold is key for generating a neural response at the envelope frequen- cies. In the aforementioned context of processing natural stimuli 9–11, this extraction is desirable, and noise is thus generally detrimental. Alternately, the envelope power may hamper the detection of other relevant low-frequency stimuli because their frequency bandwidths overlap with envelope bandwidths. This may arise, e.g., when a cortical cell partici- pating in narrowband rhythmic activity generates envelope power and simultaneously tries to detect a low-frequency stimulus. In this context, envelope extraction from this nar- rowband activity is detrimental. We go on to show how a neuron population can overcome the “noisy background” caused by the extracted envelope. It can transmit low- frequency signals in the envelope bandwidth all the while responding coherently to the narrowband input. This en- hancement of transmission, a new form of noise shaping, paradoxically arises from the addition of intrinsic uncorre- lated noise in all neurons, with subsequent population aver- aging. This is in contrast to previous mechanisms involving reciprocal feedback in a network 16 or single cell negative interspike interval correlations 17. Our results show how threshold nonlinearity and noise can gate the transmission of envelope power. The model neuron used in this study is the leaky integrate-and-fire LIF neuron 18, with dynamics, dv dt =- v  + I +  2D  t + St , 1 where v is the trans-membrane voltage,  is the membrane FIG. 1. A narrowband signal drives the input bias to a neuron near rheobase. The FI curve acts as a static transfer function, map- ping the signal to a time-varying firing rate. Under these conditions, the output firing rate is a rectified version of the input upper right. Also, the spectral power of this rate bottom right contains the same narrowband frequencies as the input, as well as the low fre- quencies of the slow time-varying envelope of this input. This en- velope is seen here using a running average of the output rate over the fast time scale thick line, upper right-hand panel. PHYSICAL REVIEW E 75, 021918 2007 1539-3755/2007/752/0219185 ©2007 The American Physical Society 021918-1