{"title": "A Model of Neural Oscillator for a Unified Submodule", "book": "Advances in Neural Information Processing Systems", "page_first": 560, "page_last": 567, "abstract": null, "full_text": "560 \n\nA MODEL OF NEURAL OSCILLATOR FOR A UNIFIED SUEt10DULE \n\nA.B.Kirillov, G.N.Borisyuk, R.M.Borisyuk, \n\nYe.I.Kovalenko, V.I.Makarenko,V.A.Chulaevsky, \n\nV.I.Kryukov \n\nResearch Computer Center \nUSSR Academy of Sciences \nPushchino, Moscow Region \n\n142292 \n\nUSSR \n\nAmTRACT \n\nfor \n\nA new model of a controlled neuron oscillatJOr, \nproposed earlier {Kryukov et aI, 1986} \nthe \ninterpretation of the neural activity in various \nparts of the central nervous system, may have \nimportant applications in engineering and in the \ntheory of brain functions. The oscillator has a \nits \ngood stability of the oscillation period, \nfrequency is regulated linearly in a wide \nrange \nand it can exhibit arbitrarily long oscillation \nperiods without changing the time constants of \nits elements. The latter is achieved by using \nthe critical slowdown in the dynamics arising in \na network of nonformal \nneurons \n{Kovalenko et aI, 1984, Kryukov, 1984}. By \nchanging the parameters of \nthe oscillator one \ncan obtain various functional modes which are \nnecessary to develop a model of higher brain \nfunction. \n\nexcitatory \n\nmE CECILLATOR \n\ninhibitory neuron. The \n\nOur oscillator comprises several hundreds of modelled \nexcitatory neurons (located at the 6i tes of a plane lattice) \nand one \nlatter receives output \nstgnals from all the excitatory neurons and its own output \nis transmitted via feedback to every excitatory neuron (Fig. \n1). Each excit~tory neuron is connected bilaterally with its \nfour nearest neighbours. \nEach neuron has a threshold r(t) decaying exponentially to a \n\n\fA Model of Neural Oscillator for a Unified Submodule \n\n561 \n\ne \n\ni \n\nimpulses \n\nincreases \n\ninwt. If the membrane potential exceeds \n\nvalue roo or roo (for an excitatory or inhibitory neuron). A \nGaussian noise with zero mean and standard deviation a \nis \nis \nadded to a threshold. A membrane potential of a neuron \nthe sum of input impulses decaying exponentially when \nthere \nare no \nthe \nthe \nthreshold, the neuron fires and sends \nto \nneighbouring neurons. An imWlse from excitatory neuron \nto \nthe membrane potential of the \nexcitatory one \nlatter by aee, from the excitatory to the \nby \ndecreases \naei, and from the inhibitory to the excitatory \nthe membrane potential by aie' We consider a discrete \ntime \nmodel J \nthe time step being equal to the absolute refractory \nperiod. \nWe associate a variable xi(t) with each excitatory neuron. \nIf the i-th neuron fires at step t, we take x.(t)=1; \nif it \ndoes not, then Xi (t)=O. The mean E(t)=l/N ~i (t) will be \nreferred to as the network acti vi ty, where N is \nthe number \nof excitatory neurons. \n\ninhibitory \n\n-\n\n-\n\n1 \n\nA \n\nB \n\n'f.! ----- ----\n\nFigure 1. A - neuron, B - scheme of interconnections \n\nis cut \nLet us consider a situation when inhibitory feedback \nthe \noff. Then such a model exhibits a critical slowdown of \ndynamics {Kovalenko et al, 1984, Kryukov, 1984}. Namely, \nif \nthe interconnections and parameters of neurons are chosen \nappropriately , initial pattern of activated neurons has an \nunusually long lifetime as compared with the time of membrane \npotential decay. In this mode R(t) is slowly \nincreasing and \n\n\f562 \n\nKirillov, et al \n\ncauses the inhibitory neuron to fire. \nNow J if we tum on the negative \nimpulse \nfrom inhibitory neuron sharply decreases membrane potentials \nof excitatory neurons. As a a consequence, K( t) falls down \nand process starts from the beginning. \nWe studied this oscillator by means of simulation model. \nThere are 400 excitatory neurons \nlattice) and one \ninhibitory neuron in our model. \n\nfeedback, outPUt \n\n(20*20 \n\nTHE MAIN PKFml'IHS OF THE &o the value of Tst is equal to the stimulation period \ntat' The dependence between &0 and tat is close to a \nlinear \none (Fig, 4B). The usual relaxation oscillator also exibitB \na linear dependence between &0 and tat' At the same time, we \ndid not find \nresonance phenomena \nessential to a linear oscillator, \n\nin our oscillator any \n\nsuch \n\n'1'HK NE'l1\u00ab)BK WITH INTKBNAL R>ISE \n\nIn a further development of the neural oscillator we \nto ooild a model \nbiological counterpart. To \nstructure of \ncorrectly the noise component of the i.np.lt signal coming \nan excitatory neuron, \n\ntried \nthe \nthe \ninterconnections and tried to define more \nto \nthe model described above we \n\nthat will be more adequate \n\nto \nchanged \n\nthis end, we \n\nIn \n\n\fA Model of Neural Oscillator for a Unified Submodule \n\n565 \n\nfrom \n\nits \n\nthe sum of \n\ninputs \n\nneurons \n\nfrom distant \n\ninformation coming \n\nimitated \nby \nindependent Gaussian noise. Here we used real noise produced \nby the network. \nIn order to simulate this internal noise, we randomly choose \n16 distant neighbours for every exi tatory neuron. Then we \nassume that the network elements are adjusted to work \nin a \ncertain noise environment. This means that a ' mean' internal \nnoise would provide conditions for the neuron to be the most \nsensitive for the \nnearest \nnelghbors . \nSo. for every neuron i we calculate the sum k. =&c . (t), where \nsummation is over all distant nelghbors of this neuron, and \ninternal noise k=1/N Lk.. The \ncompare it with the mean \ninternal noise for the neuron i now is ni=C(ki-k), where C>O \nis a constant. \nWe choose model parameters in such a way \nthe noise \ncomponent is of the order of several percent of the membrane \npotential. Nevertheless, the network exhibits in this case a \ndramatic increase of the \nlifetime of initial pattern of \nactivated neurons, as compared with \nthe network with \nindependent Gaussian noise. A range of parameters, for which \nthis slowdown of \nalso \nconsiderably irtCreased. Hence, \nlonger perioos and better \nperioo stability could be obtained for our generator if we \nuse internal noise. \n\nthe dynamics \n\nis observed, \n\nthat \n\n1 \n\nJ \n\n1 \n\nis \n\nthe total activity of \n\nsmall System constituted of \n\nTHE CHAIN OF THREE SUBMODULES: A MODEL OF COLUMN OSCILLATOR \nNow we consider a \nthree \nconnected consecutively \noscillator submodules, A, B and C, \nso that submodule A can transmit excitation to submodule B, \nB to C, and C to A. The excitation can only be \ntransmitted \nreaches its \nwhen \nthe submodule \nthreshold level, \ninhibitory \nthe \nneuron fires. After the inhibitory neuron has fired, \nactivity of its submodule is set to be small enough for \nthe \nsubmodule not to be active with large probability until the \nexcitation from another submodule comes. Therefore, we \nexpect A, B and C to work consecutively. \nin our \nsimulation experiments we observed such behavior of the \n\nthe corresponding \n\ni.e. when \n\nIn fact, \n\n\f566 \n\nKirillov, et al \n\nA \n\nSeT) \n\n20 \n\nT \n\n35 \n\no '--------\n12 \n\n10 \n\n15 L..-___ ___ _ \n12 \n\n10 \n\nFigure 5. Chain of three sutmodules. Period of \n\noscillations (A) and its standard deviation (B) vs. \n\nnoise amplitude \n\nthat 0.5 corresponds approximately to \n\nclosed chain of 3 basic submodules. The activity of \nthe \nthe \nwhole system is nearly periodic. Figure 5A displays \nperiod T vs. the noise amplitude a. The scale of a is chosen \nresting \nso \npotential. An interesting feature of the chain is that \nthe \nis small \nstandard deviation SeT) of the period \nenough, even for the oscillator of relatively small size. \nto square 10*10 \nin Fig. 5 correspond \nThe upper \nlines \nnetwork, middle -\nto 9*9, lower - to 8*8 one. One can see \nthat the loss of 36 percent of elements only causes a \nreduction of \nof \nstability. \n\nrange without \n\nthe working \n\n(Fig. 5B) \n\nloss \n\nthe \n\nthe \n\nCXHUJSI~ \n\nus \n\nthe same oscillator \n\nThough we have not considered all the interesting modes of \nthe oscillator, we believe that, owing to the phenomenon of \nmetastability, \nexhibits different \nbehaviour under slightly different threshold parameters and \nthe same and/or different inPuts. \nLet \nthe most \npossibilities of \nobtained from our results. \n1.Pacemaker with the frequency regulated in a wide range and \nwith a high period stability, as compared with \nthe neuron \n(Fig. 313). \n2. Integrator (input=threshold, output=phase) with a wide \n\nthe oscillator, which can be \n\nfunctional \neasily \n\ninteresting \n\nenumerate \n\n\fA Model of Neural Oscillator for a Unified Submodule \n\n567 \n\nrange of linear regulation (see Fig. 3A). \n3.Generator of damped oscillations (for discontinuous inPut). \n4. Delay device controlled by an external signal. \n5.Phase comparator (see Fig. 4A). \nWe have already used these functions for the \ninterPretation \nof electrical activity of several functionally different \nneural structures {Kryukov et aI, 1986}. The other functions \nwill be used in a system model of attention {Kryukov, 1989} \npresented in this volume. All these considerations \njustify \nthe name of our neural oscillator - a unified submodule for \na ' resonance' neurocomputer. \n\nReferences \n\nE. I. Kovalenko, G. N. Borisyuk, R. M. Borisyuk, A. B. \n\nI . Kryukov. \n\nKirillov, V . \na \nmetastable state. II. S1IWlation model, Cybernetics and \nSystems Research3 2, R. Trappl \n(ed.), Elsevier, pp. \n266-270 (1984) \n\nShort-tenn memory \n\nas \n\nV. I. Kryukov. Short-tenn memory as a metastable state. \nI. Master equation approach, Cybernetics and Systems \nResearch 3 2, R. Trappl \n(ed.), Elsevier, pp. 261-265 \n(1984) \nI. Kryukov. \n(1989) (in this volume). \n\n\"Neurolocator\" , a model \n\nattention \n\nof \n\nV. \n\nV. I. Kryukov, G. N. Borisyuk, R. M. Borisyuk, A. B. \nKirillov, Ye. I. Kovalenko. The Metastable and Unstable \nStates in the Brain (in Russian), Pushchino, Acad. Sci. \nUSSR (1986) (to appear in Stochastic Cellular Syste.ms: \nErgodici tY3 l1emory3 Morphogenesis, Manchester University \nPress, 1989). \n\n\f", "award": [], "sourceid": 120, "authors": [{"given_name": "Alexandr", "family_name": "Kirillov", "institution": null}, {"given_name": "G.", "family_name": "Borisyuk", "institution": null}, {"given_name": "R.", "family_name": "Borisyuk", "institution": null}, {"given_name": "Ye.", "family_name": "Kovalenko", "institution": null}, {"given_name": "V.", "family_name": "Makarenko", "institution": null}, {"given_name": "V.", "family_name": "Chulaevsky", "institution": null}, {"given_name": "V.", "family_name": "Kryukov", "institution": null}]}