White Noise Analysis of Coupled Linear-Nonlinear Systems

Abstract

Abstract. We present an asymptotic analysis of two coupled linear-nonlinear systems. Through measuring first and second input-output statistics of the systems in response to white noise input, one can completely characterize the systems and their coupling. The proposed model is similar to a widely used phenomenological model of neurons in response to sensory stimulation and may be used to help characterize neural circuitry in sensory brain regions.Key words. neural networks, correlations, Weiner analysis, white noise AMS subject classification. 92C201. Introduction. Most electrophysiology data from intact mammalian brains is recorded using an extracellular electrode which remains outside neurons. When the electrode is positioned near a neuron, it can record the neuron's output events, called spikes, because spike magnitudes are sufficiently large. The internal state of a neuron, including small fluctuations in response to its inputs, cannot be measured.When only output spikes are measurable, one cannot directly measure the effect of a connection from one neuron to another. If neuron 1 is connected to neuron 2, then an output spike of neuron 1 will perturb the internal state of neuron 2. If the internal state cannot be measured, this perturbation can be inferred only via its effect on the spike times of neuron 2. In general, the spike times of a neuron will be a function of many inputs coming from many other neurons. This complexity makes reliable inferences on the structure of neuron circuits from spike time data a formidable challenge.Explicit mathematical models may lead to tools that can address this challenge. Through model analysis, one may develop methods to infer aspects of network structure from spike times, subject to the validity of the underlying model. In this paper, we derive a method to reconstruct the connectivity between two isolated neurons based on a simple linear-nonlinear model (see below) of neural response to white noise. Although this model greatly simplifies the reality of the brain's neural networks, the results from this analysis can be used to analyze neurophysiology data provided that they are interpreted within the limitations of the model [12].Numerous researchers have used white noise analysis to describe the response of neurons to a stimulus. The most common use of white noise analysis has been to analyze the response properties of single neurons [11,4,5,8,9,2,3,16,18,7,6]. Recently, researchers have begun to apply the techniques of white noise analysis to simultaneous measurements of multiple neurons [15,1,19], although without explicitly modeling neural connectivity. In Ref.[12], we showed how, in white noise experiments, interpretation of spike time data is especially difficult because standard correlation measures confound stimulus and connectivity effects. We demonstrated correlation measures that remove the stimulus effects based on the linear-nonlinear model.In this paper, we present the asymptotic analysis of the linear-nonlinear model that underlies the correlation measur...

Revealing Pairwise Coupling in Linear-Nonlinear Networks

“…We base our analysis on a model network of linearnonlinear neurons that builds on the models we have presented previously [12,10,11,9]. Let n be the (presumably unknown) number of neurons in the network.…”

“…Because we assume a second-order approximation in coupling strengthW , it turns out that a first-order approximation inW is sufficient for the effective single-neuron parameters. 4 From a trivial generalization of the calculation in Appendix A.1 of [11], we can average the network model (2.1) over all spikes before time i to conclude that the probability of a spike at time i is…”

“…Since in practice, we will have much smaller datasets, we must reduce the bias in estimations from finite datasets using a procedure such as that outlined in [12,11,8]. We do not address such bias reduction here.…”

“…We have a total of 2M × 2 = 4M parameters for 6M equations. 11 The system is actually overdetermined. Moreover, due to the weak coupling assumption of section 2.2, the system is linear inÛ k ,W k 21 , andW −k 12 .…”

“…We repeat the calculations of Appendices A and B of [11], computing terms only up to the modified second-order approximation inW , described above in section 2.2. To simplify the notation, we defineW k pq = 0 for k ≤ 0.…”