[Comp-neuro] why math first (from one who went the other way)
ajmandell at charter.net
Tue Aug 12 18:38:51 CEST 2008
I'm a "newby" to this remarkable ongoing discussion. Several issues are up in the air, all of them of great interest to me; thank you.
I got personally moved by the important pedagogical discussion about which transition is easier, biology to math or math to biology.
If you really want it to be real, it is really really tough to go from classical neuroscience training to the level of abstract mathematics (not physical intuition driven plumbing differential equation mathematics). required by understanding the brain's "emergent" systems. At this level of abstraction, they often obey counter-intuitive theorems which then can be projected down (if you are lucky)to a piece of the data space. Even Feynman, who hated mathematicians arrogance, always started from there and then went to the relevant physics (read his biography). Its taken me years at math institutes, really personally friendly generous great mathematicians (Rene Thom, David Ruelle, Chris Zeeman, Mike Friedman and others) and twenty years of private study and I still haven't made it. Math courses (I don't mean computer science mathematics) should start at the same time as general biology at Universities, the freshman year. Math phobics (created by inadequate and defensive grammar and high school teachers) either have to tuck it in or go somewhere else. Certainly brain science will wind up such that "real knowledge" will be at least as abstract and remarkably counter-intuitive as quantum mechanics. Theorems from complex analysis and linear algebra on complex spaces ran way ahead of the experiments, sometimes by decades.
Nancy Kopell and group is a perhaps a too good example of someone who went the math to biology way, dragging some theorems (Hopf etc) with her. Bard served as her brilliant anchor to computational brain science in a most effective way. When they added experimental neurobiologists to their group, some of the real stuff came out. I really believe that group could be done in one head if the early and fundamental training expected it.
Forgive the ego part now, I want to call your attention to the level of (still unresolved) mathematical abstraction that can emerge from just two
coupled "neurons". There are phase and parameter space structures that we still can't understand. You might enjoy the pictures though.
A REALISTIC, MINIMAL "MIDDLE LAYER" FOR NEURAL NETWORKS
Physica D 40 (1989) 135-155
"To demonstrate some dynamical consequences of making the formal characteristic of neuronal elements and their
connectivities more realistic than those of modern neural computer algorithms, we study two multiplicatively coupled logistic
maps, each with the quadratic response characteristics of neurons with autoreceptor mechanisms, which iteratively exchange
their outputs. The dynamic phenomenology of this two-neuron system, though noninvertible, demonstrates characteristics of
both automorphisms of the real line and diffeomorphisms of the plane, with smooth parametric transitions between them. We
portray a unified parameter space containing the negative, positive, and mixed-coupled regimes, categorize the phase space
structures using characteristic exponent inequalities, explore representative parametric paths, demonstrate generic bifurcation
sequences, and report number theoretic ordering in parameter space.
We conjecture that most, if not all, "middle-layer-like" brain systems, in contrast to primary sensory and motor information
transport systems, are dominated by intrinsic dynamics of the sort demonstrated here. External input serves as a perturbation
of these already ongoing complex systems.
The intrinsic instabilities of these middle-layer dynamics return the unpredictability required by theoretical studies of real
brain function to models of neural networks. We demonstrate a variety of parameter-dependent intrinsically ordered attractor
sequences and number theoretic series which emerge from the global dynamics and exemplify the rich dynamical "machine"
language potential for specific coding by brain-like information transport and learning algorithms."
Thanks for putting up with me.
Arnold J. Mandell, M.D.
Cielo Institute, Asheville, NC
ajmandell at charter.net
Clinical Professor of Psychiatry and Human Behavior, Emory University,
Adjunct Professor of Mathematical Sciences, FAU, Boca Raton, FL
Founding Chairman and Professor Emeritus of Psychiatry (Neuroscience,
Pharmacology, Mathematics) UCSD, La Jolla, CA
MacArthur Prize Fellow Laureate in Theoretical Neuroscience
Humboldt Prize Fellow Laureate in Dynamical Systems
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