Amplification by stochastic interference

A new method is introduced to obtain a strong signal by the interference of weak signals in noisy channels. The method is based on the interference of 1/f noise from parallel channels. One realization of stochastic interference is the auditory nervous system. Stochastic interference may have broad potential applications in information transmission by parallel noisy channels. The method of stochastic interference was originally conceived for information processing in the auditory nervous system [1]. It makes use of the random fractal geometry of the spike discharge patterns [2–5] which are processed by diverging and converging information networks of the auditory system. This method is distinct from stochastic resonance [6], but when both methods are combined, a fascinating new model of transsynaptic information transfer emerges [7]. Here, we are interested in more general aspects of stochastic interference. The method can be sketched as follows. Consider an information transmission via multiple channels. Assume further that the information is coded in statistically self-similar, random [8–14] fractal patterns [15]. The idea that information is encoded in the dimensional geometry of random fractals is not entirely new [2–5]. But here, n fractal information signals (with the same dimensional parameter) are combined by logical ‘and’ operations (equivalent to the set theoretic intersection) to form a new signal. The new signal also has a fractal geometry. Its fractal dimension varies n times as strongly as the variations of the dimensional parameter of the primary signal. Thus, when multiple information channels are combined properly, arbitrary weak variations of their input signals can be amplified to arbitrary strong variations of the resulting output channel. Stochastic interference operates with 1/f β noise [16, 17], characterized by a power spectral density of SV (f ) ∝ 1/f β . This noise corresponds to a signal X(t) at time t whose graph {(t, X(t)) | tmin 6 t 6 tmax} has a random fractal geometry. The fractal (box-counting) dimension of the graph can be approximated by [18, 19] D = min { 2, E + 3 − β 2 } (1) ∗ E-mail: [email protected] 0305-4470/96/130351+04$19.50 c © 1996 IOP Publishing Ltd L351 L352 Letter to the Editor where E is the (integer) dimension of the noise. For one-dimensional noise, E = 1. White noise corresponds to β = 0, brown noise corresponds to β = 2, whereas systems showing 1/f noise operate at approximately β = 0.8–1.2. Consider a sequence of zeros and ones which constitutes a fractal pattern. Such a random fractal of dimension D can, for instance, be recursively generated starting with a sequence of ones. First, the sequence is subdivided into k blocks of sequences of length δ symbols. Then, a fraction of 1 − exp[(D − 1) log(k)] blocks of length δ symbols is filled with zeros (instead of ones). Next, one takes the remaining pieces of the pattern containing ones and repeats the same procedure (the length of the blocks decreases by a factor of k, until one arrives at δ = 1) [19]. The fractal dimension of a random fractal signal can be understood as follows. Divide a sequence of zeros and ones again into k blocks of length δ. Count how many of these blocks contain ones at all (or, more realistically for practical applications, up to a density s). If r is the number of filled blocks, then the fractal (box-counting) dimension is given by D = log r log(1/δ) (2) independently of the scale resolution δ. The fractal dimensional measure D should be robust with respect to variations in methods of determining it. That is, it should remain the same, regardless of the method by which it is inferred. Information can be encoded by the random fractal patterns of 1/f noise, in particular by variations of the dimension parameter. More precisely, assume, for example, two source symbols s1 and s2 are encoded by (RFP stands for ‘random fractal pattern’) #(si) = { RFP with 0 6 D(RFP) < Dc if si = s1 RFP with Dc 6 D(RFP) 6 E if si = s2 (3) where Dc is a ‘critical dimension parameter’. As has been pointed out by Falconer [19], under certain ‘mild side conditions’, the intersection of two random fractals A1 and A2 which can be minimally embedded in R is again a random fractal with dimension D(A1 ∩ A2) = max{0, D(A1) + D(A2) − E}. (4) By induction, (4) generalizes to the intersection of an arbitrary number of random fractal sets. Thus, the dimension of the intersection of n random fractals A = {A1, . . . , An} is given by