Critical Branching Neural Computation, Neural Avalanches, and 1/f Scaling

Critical Branching Neural Computation, Neural Avalanches, and 1/f Scaling Christopher T. Kello (ckello@ucmerced.edu) Bryan Kerster (bkerster@ucmerced.edu) Eric Johnson (ejohnson5@ucmerced.edu) Cognitive and Information Sciences, 5200 North Lake Rd., Merced, CA 95343 USA Abstract Scaling Laws in Neural and Behavioral Activity It is now well-established that intrinsic fluctuations in human behavior tend to exhibit long-range correlations in the form of 1/f scaling. Their meaning is an ongoing matter of debate, and some researchers argue they reflect the tendency for neural and bodily systems to poise themselves near critical states. A spiking neural network model is presented that self-tunes to a critical point in terms of its spike branching ratio (i.e. critical branching). The model is shown to exhibit 1/f scaling near critical branching, as well neural avalanches, and critical branching is associated with maximal computational capacity when assessed in terms of reservoir computing. The model provides a basis for connecting neural and behavioral activity and function via criticality. Keywords: Critical branching, 1/f scaling, neural avalanche, criticality, metastability, reservoir computing. Introduction Variability is the essence of neural and behavioral activity, and this variability is what theories of cognition must ultimately account for. Some of this variability can be ascribed to effects of sensory stimulation, but much of it is intrinsic in nature (Fox, Snyder, Vincent, & Raichle, 2007). Like all biological systems, neural and behavioral systems exhibit activities that can neither be attributed to extrinsic factors, nor controlled by them. These systems are constantly at work to maintain themselves, and this work results in intrinsic variations in activities. The nature of intrinsic variability provides basic information about how components of these systems work together. Intrinsic variability is observed when experimental manipulations are minimized, e.g. when spontaneous neural activity is measured in cortical slice preparations (Beggs & Plenz, 2003), or in brain images during the wakeful resting state (Bullmore et al., 2001), or when behavioral acts are repeated with minimal variation in intentions and measurement conditions (Kello, Anderson, Holden, & Van Orden, 2008). What should one expect from system activity when components are in this “relaxed”, default state? A reasonable hypothesis is that component activities (e.g. neurons, cortical columns, brain areas, muscle groups, etc.) decouple to become relatively independent, and effectively random. If fluctuations in system activities reflect component sums in intrinsic measurement conditions, then activities should tend towards “white noise”, i.e. random samples drawn from a normal distribution. In fact this is the basic assumption of linear models with Gaussian error terms. However, numerous studies of intrinsic variability do not bear out this assumption. In many different studies of neural and behavioral activity, intrinsic variations have been reported to follow scaling laws across a wide range of scales. Scaling laws generally relate one variable as function of another raised to a power, f(X) ~ X a , where typically a < 0. Well-known examples from psychology and cognitive science include Steven’s law, Zipf’s law, scale-free semantic networks, and power laws of learning and forgetting (for review see Kello et al., 2010). Here we focus on two different scaling laws that have attracted a great deal of attention in recent years. One is a power law distribution in neural activity referred to as a “neural avalanche” (Beggs & Plenz, 2003), and the other is long-range correlated fluctuations in behavioral and neural activity, known as 1/f scaling (Kello et al., 2008). The term “neural avalanche” originally referred to bursts of neural spiking activity found in local field potentials recorded from slice preparations that are designed for observing intrinsic variations. Probability distributions of burst sizes S were found to go as P(S) ~ 1/S β , where β ~ 3/2 over a moderate range of scales. Analogous burst size distributions have also been found in EEG, MEG, and fMRI recordings (see Poil, van Ooyen, & Linkenkaer-Hansen, 1/f scaling refers to autocorrelations in time series of repeated measurements, in our case taken from neural or behavioral systems. Each measurement is correlated with previous ones, and correlations decay slowly as an inverse power of lag between measurements. In the frequency domain, this scaling relation holds between spectral power S and frequency f as S(f) ~ 1/f α , where α ~ 1 over a moderate to wide range of scales. This scaling law has been observed in local field potentials, EEG, fMRI, and a wide variety of behavioral measures of intrinsic variation, including tapping, walking, reaction times, interval estimates, and the acoustics of spoken word repetitions (see Kello et al., 2008). Criticality and Computation What do neural avalanches and 1/f scaling tell us about neural and behavioral systems? One possibility is suggested by the particular exponent values observed, because they are both predicted to occur in the intrinsic variations of systems near their critical points. Critical points occur at the transitions between phases (i.e. modes) of component interactions, and many different kinds of complex systems have been hypothesized to self-organize towards their critical points (Bak, 1996). Theoretical work has shown that