Abstract
Neuronal membrane and synapse polarities have been attracting considerable interest in recent years. Certain functional roles for such polarities have been suggested, yet, they have largely remained a subject for speculation and debate. Here, we note that neural circuit polarity codes, defined as sets of polarity permutations, divide into primal-size circuit polarity subcodes, which, sharing certain connectivity attributes, are called categories. Two long-debated, seemingly competing paradigms of neuronal self-feedback, namely, axonal discharge and synaptic mediation, are shown to jointly define the distinction between these categories. However, as the second paradigm contains the first, it is mathematically sufficient for complete specification of all categories. The analysis of primal-size circuit polarity categories is found to reveal, explain and extend experimentally observed cortical information capacity values termed “magical numbers”, associated with “working memory”. While these have been previously argued on grounds of psychological experiments, here they are further supported on analytic grounds by the so-called Hebbian memory paradigm. The information dimensionality associated with these capacities is found to be a consequence of prime factorization of composite circuit polarity code sizes. Different categories of circuit polarity, identical in size and neuronal parameters, are shown to generate different firing rate dynamics.
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References
Abbott LF (1994) Decoding neuronal firing and modeling neural networks. Q Rev Biophys 27:291–331
Amit DJ, Gutfreund H, Sompolinsky H (1987) Statistical mechanics of neural networks near saturation. Ann Phys 173(1):30–67
Arimura N, Kaibuchi K (2005) Key regulators in neuronal polarity. Neuron 48(6):881–884
Atwood HL, Wojtowicz JM (1999) Silent synapses in neural plasticity: current evidence. Learn Mem 6:542–571
Awodey S (2010) Category theory, 2nd edn. Oxford logic guides. 49. Oxford University Press, Oxford
Baram Y (2018) Circuit polarity effect of cortical connectivity, activity, and memory. Neural Comput 30(11):3037–3071
Bengtsson I, Życzkowski K (2017) Geometry of quantum states: an introduction to quantum entanglement, 2nd edn. Cambridge University Press, Cambridge, pp 313–354
Bick C, Rabinovich MI (2009) Dynamical origin of the effective storage capacity in the brain’s working memory. Phys Rev Lett 103:218101-1–218101-4
Bienenstock EL, Cooper LN, Munro PW (1982) Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. J Neurosci 2:32–48
Carandini M, Ferster D (2000) Membrane potential and firing rate in cat primary visual cortex. J Neurosci 20:470–484
Carlsson A, Lindquist MA (1963) Effect of chlorpromazine and haloperidol on formation of 3-methoxytyramine and normetanephrine in mouse brain. Acta Pharmacol Toxicol 20:140–144
Castellani GC, Quinlan EM, Cooper LN, Shouval HZ (2001) A biophysical model of bidirectional synaptic plasticity: dependence on AMPA and NMDA receptors. Proc Natl Acad Sci USA 98(22):12772–12777
Chase WG, Ericsson KA (1982) Skill and working memory. In: Bower GH (ed) Psychology of learning and motivation, vol 16. Academic Press, Cambridge, pp 1–58
Colman AM (2008) A dictionary of psychology. Oxford University Press, Oxford
Connor JA, Stevens CF (1971) Voltage clamp studies of a transient outward membrane current in gastropod neural somata. J Physiol 213:21–30
Cooper LN, Intrator N, Blais BS, Shouval HZ (2004) Theory of cortical plasticity. World Scientific, Hackensack
Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 2nd edn. MIT Press, Cambridge, pp 232–236
Cowan N (1995) Attention and memory: an integrated framework. Oxford University Press, Oxford
Cowan N (2001) The magical number 4 in short-term memory: a reconsideration of mental storage capacity. Behav Brain Sci 24(1):87–114
Dayan P, Abbott LF (2001) Theoretical neuroscience. MIT Press, Cambridge, MA
du Sautoy M (2003) The music of the primes: searching to solve the greatest mystery in mathematics. Harper Collins, New York
Furstenberg H (1955) On the infinitude of primes. Am Math Month 62(5):353
Gerstner W (1995) Time structure of the activity in neural network models. Phys Rev E 51:738–758
Granit R, Kernell D, Shortess GK (1963) Quantitative aspects of repetitive firing of mammalian motoneurons caused by injected currents. J Physiol 168:911–931
Groves PM, Wilson CJ, Young SJ, Rebec GV (1975) Self-inhibition by dopaminergic neurons: an alternative to the “neuronal feedback loop” hypothesis for the mode of action of certain psychotropic drugs. Science 190:522–528
Hafting T, Fyhn M, Molden S, Moser MB, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex”. Nature 436(7052):801–806
Hebb DO (1949) The organization of behavior: a neuropsychological theory. Wiley, New York
Hodgkin A, Huxley AA (1952) Quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544
Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Nat Acad Sci USA 79:2554–2558
Intrator N, Cooper LN (1992) Objective function formulation of the BCM theory of visual cortical plasticity: statistical connections, stability conditions. Neural Netw 5:3–17
Kimata T, Tanizawa Y, Can Y et al (2012) Synaptic polarity depends on phosphatidylinositol signaling regulated by myo-inositol monophosphatase in Caenorhabditis elegans. Genetics 191(2):509–521
Kirtland J (2001) Identification numbers and check digit schemes. Classroom Resource Materials. 18. Mathematical Association of America, Washington, pp 43–44
Křížek M, Luca F, Somer L (2001) Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics. Springer, New York, pp 1–2
Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation. J Physiol Pathol Gen 9:620–635
Lehmer E (1936) On the magnitude of the coefficients of the cyclotomic polynomial. Bull Am Math Soc 42(6):389–392
Matsumoto M, Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans Model Comput Simul 8(1):3–30
McCulloch WS, Pitts W (1943) A logical calculus of the idea immanent in nervous activity. Bullet Math Biophys 5:115–133
Melnick IV (1994 Rus, 2010 Eng) Electrically silent neurons in the substantia gelatinosa of the rat spinal cord. Fiziol Zh 56(5):34–9
Miller GA (1956) The magical number seven, plus or minus two: some Limits on our Capacity for Processing Information. Psychol Rev 63:81–97
Miller GL (1976) Riemann’s Hypothesis and Tests for Primality. J Comput Syst Sci 13(3):300–317
Miller KD, Fumarola F (2012) Mathematical equivalence of two common forms of firing rate models of neural networks. Neural Comput 24:25–31
Neumann N (2010) The mind of the mnemonists: an MEG and neuropsychological study of autistic memory savants. Behav Brain Res 215(1):114–121
O’Keefe J, Dostrovsky J (1971) The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res 34(1):171–175
Peterson I (1999) The Return of Zeta, MAA on Line. Math Soc Amer. https://static1.squarespace.com/static/54c161ffe4b063fc8ab03446/t/54cab277e4b042fd7653bec5/1422570103763/The+Return+of+Zeta.pdf. Accessed 19 Aug 2019
Pribram KH, Miller GA, Galanter E (1960) Plans and the structure of behavior. Holt, Rinehart and Winston, New York, p 65
Qwakenaak H, Sivan R (1972) Linear optimal control systems. Wiley, New York
Rabin MO (1980) Probabilistic algorithm for testing primality. J Number Theory 12(1):128–138
Ribenboim P (2017) Prime numbers, friends who give problems. World Scientific, Hackensack
Smith TC, Jahr CE (2002) Self-inhibition of olfactory bulb neurons. Nat Neurosci 5:760–766
Tanizawa Y, Kuhara A, Inada H et al (2006) Inositol monophosphatase regulates localization of synaptic components and behavior in the mature nervous system of C. elegans. Genes Dev 20(23):3296–3310
Williams KS, Simon C (1995) The ecology, behavior, and evolution of periodical cicadas. Annu Rev Entomol 40:269–295
Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24
Yaro C, Ward J (2007) Searching for Shereshevskii: what is superior about the memory of synaesthetes? Q J Exp Psychol 60(5):681–695
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Baram, Y. Primal categories of neural polarity codes. Cogn Neurodyn 14, 125–135 (2020). https://doi.org/10.1007/s11571-019-09552-x
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DOI: https://doi.org/10.1007/s11571-019-09552-x