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Wild Random Brain Noise
Lawrence M. Ward
Dynamical Cognitive Science
We previously noted that for Benoit Mandelbrot, there is a certain kind of noise pattern, 1/f, that he classifies as "wild randomness." It is wild, because "anomalous" erratic events in fact are frequent yet still totally unpredictable. We now discuss research that suggests our brains function in accordance with such wild 1/f noise.
Ch 15
Colored Noise
[1/f Noise has a -1 line (down one, over one).]
Noise with this spectral "shape" is called "pink noise" because it is dominated by relatively low frequencies, but has some higher frequencies in it — just as light that appears pink to human eyes is dominated by low-frequency (long-wavelength) photons in it. As the supposed signature of a very interesting type of dynamical system called a "complex system," pink noise has proved notoriously difficult to explain. I will have more to say about complex systems, but for now it suffices to point out that, wherever pink noise appears, it is reasonable to look for other exotic phenomena, even in some cases the one called "chaos." (126b-c)
Ch. 16
1/f Noise in the Brain
The neuron's instantaneous rate of firing (i.e., of generating action potentials) is the dimension usually considered to encode the information, but more recently the time interval between successive pulses has also been suggested as an encoding dimension (e.g, Singer, W., 1999, "Neuronal Synchrony: A versatile code for the definition of relations?" Neuron, 24, 49-65.) Also, synchrony between the firing rates of several neurons in a network is supposed to be important, possibly as a solution to the "binding problem" of cognitive neuroscience. Of course because it will be affected by the ever present noise, the transmission of information by neurons cannot be completely precise. The noise will manifest itself in changes in the rate with which action potentials are generated by the neuron, and possibly in changes in the speed with which each one propagates down the axon. The noise arises from sources such as fluctuations in the ionic concentrations inside and outside the axon and other "channel noise" (White, Rubenstein and Kay, 2000, "Channel noise in neurons." Trends in Neuroscience, 21, 131-137.), fluctuations in the temperature of the surrounding fluids, and the decaying effects of previous action potentials received by the neuron. Some channel noise has been shown to be 1/f, probably arising from the vibration of hydrocarbon chains in the lipids in the nerve cell membrane affecting conductance of potassium ions through the membrane (Lundström and McQueen, 1974, "A proposed 1/f noise mechanism in nerve cell membranes," Journal of Theoretical Biology, 45, 405-409). The effect of previous action potentials is perhaps the most interesting in the present context, however, because it resembles a memory or relaxation process source for 1/f noise. After each action potential is generated, the neuron experiences an absolute refractory period of about 1 msec, during which no new action potentials can be generated, and an exponentially decreasing relative refractory period of an additional several msec, during which the probability of generating new action potential gradually increases. Thus the generation of a particular action potential affects the probability of generating another one for quite some time afterwards; the neuron "remembers" its previous activity and that memory is combined with current inputs to yield current activity. (145-146c)Musha (1981, "1/f fluctuations in biological systems," In P.H.E. Meijer, R.D. Mountain, and R.J. Soulen, Jr., eds., Sixth International Conference on Noise in Physical Systems, 1473-146. Washington, DC: U.S. Department of Commerce and National Bureau of Standards) did some provocative experiments on the effects of previous action potentials on the time encoding of information by the giant axons of the squid, the easiest of all axons to work with, extensively studied since they were discovered by J.Z. Young. First, by exciting successive actions potentials with an electrical pulse, Musha showed that the refractory period also decreases the speed of transmission of the action potential in the axon, dropping from near 25m/sec for the first excitation to near 10 m/sec for later ones. Clearly, this would affect the encoding of information, whether the instantaneous rate of firing or the inter-action potential interval were the encoding dimension. After stimulating the axon with sequences of random electrical pulses (white noise), Musha recorded time series of the fluctuations in the time density (the inverse of transmission speed) of action potentials traveling down tthe axon. The power spectra for several such time series of density fluctuations are show in figure 17.1
along with the power spectrum of the electrical pulses that stimulated the action potentials (at the bottom of the graph). Below about 10 Hz, the action potential power spectra are approximately 1/f, whereas the spectrum of the stimulating pulses in that frequency region is white (flat). Thus the neurons, the basic building blocks of the brain, themselves display 1/f noise in the foundational mechanism of information transmission, the conduction of action potentials along the axon. Interesting, Musha and Higuchi (Muscha, 1981) had demonstrated the resemblance of the fluctuations of action potential speed to fluctuations of the speed of automobiles in traffic. The 1/f fluctuations in the traffic model are attributed to the "bunching" of the cars as they are forced to slow down by their proximity to other cars, a property of a more general statistical queueing theory approach to 1/f noise first described by Bell (1960, Electrical Noise. London: Van Nostrand.) (146c-147c)From the perspective of physics, the brain is a system with strong interactions of many degrees of freedom. It consists of perhaps 100 billion neurons, each with up to 10, 000 connections to other neurons. These connections form hierarchical (and nonhierarchical) groups, from small groups of tens of neurons to large groups consisting of entire sensory, cognitive, or motor processing areas, such as the visual cortex, with many millions of neurons. In physics, such systems are usually described by a "similarity regime," in which similar behavior is observed at several scales. Under certain conditions, a 1/f power spectrum of temporal fluctuations can arise from such a similarity regime. Novikov et al. (Novikov E., A. Novikov, Shannonhoff-Khalsa, Schwartz, and Wright, 1997, "Scale-similar activity in the brain," Physical Review E, 56, R2387-R2389,) recorded the magnetoencephalogram to establish the existence of such a regime in the human brain. (148c)
[Chart shows peaks of specific brain wave frequencies.]
More interesting are the average slopes of these power spectra, represented by the straight lines in the graphs. These lines have slopes of -1.03 and -1.19, respectively over the range 0.4 to 40 Hz, very near the -1 expected for 1/f fluctuations. Power spectra for other sensors were very similar. Moreover, taking the difference between pairs of sensors usually eliminated the peaks and yielded even more stable 1/f spectra. ... Further analyses of the data established that the scale similarity implied by the 1/f spectra was relatively "local," meaning that it extended over only limited brain areas, probably related to the shared function of those areas. Thus the spontaneous activity of functionally related chunks of the human brain exhibits 1/f noise. This finding supports the assumptions made in chapter 16 regarding the origin of the 1/f spectra in human cognition. (150 a-b)Not only is the evoked activity of the brain 1/f, but also the noise in which the evoked activity is embedded — and which possibly characterizes the process generating that evoked activity — is also 1/f. It is clear that the human brain is characterized by 1/f noise in many activity regimes, from ion flow in neurons to activity evoked by external stimuli. The challenge now is to discover the functional implications of this fact. (153c)
Ward, Lawrence M. Dynamical Cognitive Science. London: MIT Press, 2002.
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