{"title": "Bifurcation Analysis of a Silicon Neuron", "book": "Advances in Neural Information Processing Systems", "page_first": 731, "page_last": 737, "abstract": null, "full_text": "Bifurcation Analysis of a Silicon Neuron \n\nGirish N. Patel] , Gennady s. Cymbalyuk2,3, \n\nRonald L. Calabrese2, and Stephen P. DeWeerth1 \n\nlSchool of Electrical and Computer Engineering \n\nGeorgia Institute of Technology \n\nAtlanta, Ga. 30332-0250 \n\n{girish.patel, steve.deweerth} @ece.gatech.edu \n\n2Department of Biology \n\nEmory University \n\n1510 Clifton Road, Atlanta, GA 30322 \n{gcym, rcalabre}@biology.emory.edu \n\n3Institute of Mathematical Problems in Biology RAS \nPushchino, Moscow Region, Russia 142292 (on leave) \n\nAbstract \n\nWe have developed a VLSI silicon neuron and a corresponding mathe(cid:173)\nmatical model that is a two state-variable system. We describe the cir(cid:173)\ncuit implementation and compare the behaviors observed in the silicon \nneuron and the mathematical model. We also perform bifurcation analy(cid:173)\nsis of the mathematical model by varying the externally applied current \nand show that the behaviors exhibited by the silicon neuron under corre(cid:173)\nsponding conditions are in good agreement to those predicted by the \nbifurcation analysis. \n\n1 Introduction \n\nThe use of hardware models to understand dynamical behaviors in biological systems is \nan approach that has a long and fruitful history [1 ][2]. The implementation in silicon of \noscillatory neural networks that model rhythmic motor-pattern generation in animals is \none recent addition to these modeling efforts [3][4]. The oscillatory patterns generated by \nthese systems result from intrinsic membrane properties of individual neurons and their \nsynaptic interactions within the network [5]. As the complexity of these oscillatory sili(cid:173)\ncon systems increases, effective mathematical analysis becomes increasingly more impor(cid:173)\ntant to our understanding their behavior. However, the nonlinear dynamical behaviors of \nthe model neurons and the large-scale interconnectivity among these neurons makes it \nvery difficult to analyze theoretically the behavior of the resulting very large-scale inte(cid:173)\ngrated (VLSI) systems. Thus, it is important to first identify methods for modeling the \nmodel neurons that underlie these oscillatory systems. \n\nSeveral simplified neuronal models have been used in the mathematical simulations of \npattern generating networks [6][7][8] . In this paper, we describe the implementation of a \n\n\f732 \n\nG. N Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. DeWeerth \n\ntwo-state-variable silicon neuron that has been used effectively to develop oscillatory net(cid:173)\nworks [9][10]. We then derive a mathematical model of this implementation and analyze \nthe neuron and the model using nonlinear dynamical techniques including bifurcation \nanalysis [11]. Finally, we compare the experimental data derived from the silicon neuron \nto that obtained from the mathematical model. \n\n2 The silicon model neuron \n\nThe schematic for our silicon model neuron is shown in Figure 1. This silicon neuron is \ninspired by the two-state, Morris-Lecar neuron model [12][ 13]. Transistor M I ' analo(cid:173)\ngous to the voltage-gated calcium channel in the Morris-Lecar model, provides an instan(cid:173)\ntaneous inward current that raises the membrane potential towards V High when the \nmembrane is depolarized. Transistor M2 ' analogous to the voltage-gated potassium chan(cid:173)\nnel in the Morris-Lecar model, provides a delayed outward current that lowers the mem(cid:173)\nbrane potential toward V Low when the membrane is depolarized. V H and V L are \nanalogous to the half-activation voltages for the inward and outward currents, respec(cid:173)\ntively. The voltages across C I and C2 are the state variables representing the membrane \npotential, V, and the slow \"activation\" variable of the outward current, W, respectively. \nThe W -nullcline represents its steady-state activation curve. Unlike the Morris-Lecar \nmodel, our silicon neuron model does not possess a leak current. \n\nUsing current conservation at node V, the net current charging CI is given by \n\nwhere iH and iL are the output currents of a differential pair circuit, and a p and aN \ndescribe the ohmic effects of transistors M J and M2, respectively. The net current into C2 \nis given by \n\n(1) \n\nwhere ix is the output current of the OTA, and ~p and ~N account for ohmic effects of \nthe pull-up and the pull-down transistors inside the OTA. \n\n(2) \n\nVHigh \n\nOTA \n\nv \n\nw \n\n'------<>--- V Low \n\nFigure 1: Circuit diagram of the silicon neuron. The circuit incorporates analog building \nblocks including two differential pair circuits composed of a bias current, IB H, and \ntransistors M4-Ms, and a bias current, IBL, and transistors M6-M7' and a single follower(cid:173)\nintegrator circuit composed of an operational transconductance amplifier (OTA), Xl in the \nconfiguration shown and a load capacitor, C2. The response of the follower-integrator \ncircuit is similar to a first-order low-pass filter. \n\n\fBifurcation Analysis of a Silicon Neuron \n\n733 \n\nThe output currents of the differential-pair and an OTA circuits, derived by using sub(cid:173)\nthreshold transistor equations [2], are a Fenni function and a hyperbolic-tangent func(cid:173)\ntion, respectively [2]. Substituting these functions for iH, iL, and ix in (1) and (2) yields \n\n. \n\nC 1 V = Iexta p + IBH \n\ne \n\nK(V-YH) / U T \n\nK(W - YL) / UT \n\ne \n\nK(V _ YH) / UT a p -\n\nIBL \n\nK(W _ YL) / U T aN \n\nl+e \n\nl+e \n\nwhere \n\na p = \n\nI\n\nv - V High / UT \n\n-e \n\naN = 1 - e \n\nY Low - V I UT \n\n~N = 1- e \n\n- W / UT \n\n(3) \n\n(4) \n\nU T is the thennal voltage, V dd is the supply voltage, and K is a fabrication dependent \nparameter. The tenns a p and aN limit the range of V to within V High and V Low' and the \nterms ~p and ~N limit the range of W to within the supply rails (Vdd and Gnd). \n\nIn order to compare the model to the experimental results, we needed to determine val(cid:173)\nues for all of the model parameters. V Hi!\\h' V Low' V H' V L ' and V dd were directly mea(cid:173)\nsured in experiments. The parameters IBH and IBL were measured by voltage-clamp \nexperiments performed on the silicon neuron. At room temperature, UT ::::: 0.025 volts. \nThe value of K ::::: 0.65 was estimated by measuring the slope of the steady-state activa(cid:173)\ntion curve of inward current. Because W was implemented as an inaccessible node, IT \ncould only be estimated. Based on the circuit design, we can assume that the bias cur(cid:173)\nrents IT and IBH are of the same order of magnitude. We choose IT::::: 2.2 nA to fit the \nbifurcation diagram (see Figure 3). Cl and C2, which are assumed to be identical accord(cid:173)\ning to the physical design, are time scaling parameters in the model. We choose their val(cid:173)\nues (Cl = C2 = 28 pF) to fit frequency dependence on lext (see Figure 4). \n\n3 Bifurcation analysis \n\nThe silicon neuron and the mathematical model! described by (3) demonstrate various \ndynamical behaviors under different parametric conditions. In particular, stable oscilla(cid:173)\ntions and steady-state equilibria are observed for different values of the externally applied \ncurrent, I ext . We focused our analysis on the influence of I ext on the neuron behavior for \ntwo reasons: (i) it provides insight about effects of synaptic currents, and (ii) it allows \ncomparison with neurophysiological experiments in which polarizing current is used as a \nprimary control parameter. The main results of this work are presented as the comparison \nbetween the mathematical models and the experimental data represented as bifurcation \ndiagrams and frequency dependencies. \nThe null clines described by (3) and for lext = 32 nA are shown in Figure 2A. In the \nregime that we operate the circuit, the W -null cline is an almost-linear curve and the V(cid:173)\nnullcline is an N-shaped curve. From (3), it can be seen that when IBH + lext > IBL the \nnullclines cross at (V, W)::::: (VHigh, VHigh ) and the system has high voltage (about 5 \nvolts) steady-state equilibrium. Similarly, for Iext close to zero, the system has one stable \nequilibrium point close to (V, W) ::::: (V Low' V Low). \n\n!The parameters used throughout the analyses of the model are V Low = 0 V , \nV High = 5 V, V L = V H = 2.5 V, I BH = 6.5 nA , I BL = 42 nA, IT = 2.2 nA , \nV dd = 5 V, Vt = 0.025 mV, and K = 0.65. \n\n\fG. N Patel, G. S. Cymba/yule, R. L. Calabrese and S. P. De Weerth \n\nW-nullcline \n\n/ \n\n/ @ \n\n/ \n\nV-nullcline \n\n734 \n\nA \n\n2.85 \n\n2.8 \n\n2.75 \n\n~ 2.7 \n(5 \n> \n.....,.. \n~ 2.65 \n\n2.6 \n\n2.55 \n\n2.5 \n\n2.45~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~1 \n\nI \n\no \n\n2 \n\n3 \n\n4 \n\n5 \n\nV (volts) \n\nB \n\n3.2 \n\n3 \n\n2.8 -In -l2.6 \n\n> \n\n2.4 \n\n2.2 \n\n2 \n\n0 \n\n5 \n\n10 \n\n15 \n20 \ntime (msec) \n\n25 \n\n30 \n\n35 \n\nFigure 2: Nullclines and trajectories in the model of the silicon neuron for lex! = 32 nA. \nThe system exhibits a stable limit-cycle (filled circles), an unstable limit-cycle (unfilled \ncircles), and stable equilibrium point. Unstable limit-cycle separates the basins of \nattraction of the stable limit-cycle and stable equilibrium point. Thus, trajectories initiated \nwithin the area bounded by the unstable limit-cycle approach the stable equilibrium point \n(solid line in A's inset, and \"x's\" in B). Trajectories initiated outside the unstable limit(cid:173)\ncycle approach the stable limit-cycle . In A, the inset shows an expansion at the \nintersection of the V - and W -nullclines. \n\n\fBifurcation Analysis of a Silicon Neuron \n\n735 \n\nA \n\n5 \n\n4 \n\nExperimental data \n........................... ~ \n\u2022 \n\u2022 \n\u2022 \n\u2022 \n.. \n\u2022 \n\u2022 \n\u2022 \n\u2022 \n.. \u2022 \nI \n\u2022 \n\n\u2022 \n\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 \n\n\u2022 \n\u2022 \u2022 \n\u2022 \u2022 \n\nx xxxx \u2022\u2022 x \n\n\",>0< \n\nx \n\nx \n\n~ \n\n~ \n\n003 \n.-\n0 \n2:-\n>2 x \n\n>t#* xxxx \n\n1 \n\n0 \n\nx \n\n0 \n\n10 \n\n20 \nlext \n\n30 \n\n(nAmps) \n\n40 \n\n50 \n\nB \n\nModeling data \n\n4 \n\n5 \n\n( \n\u2022 \n\u2022 \n\n....................... -. \n. \n\u2022 \n\u2022 \n\u2022 \n0 rt---------; \n,J \n> ->2 \n\n003 \n.-\n\n\u2022 \n\n. \n'-....................... \n\n\u2022 \n\u2022 \n~ \n\n1 \n\n0 \n\n0 \n\n10 \n\n20 \n30 \nI ext (nAmps) \n\n40 \n\n50 \n\nFigure 3: Bifurcation diagrams of the hardware implementation (A) and of the \nmathematical model (B) under variation of the externally applied current. In A, the steady(cid:173)\nstate equilibrium potential of V is denoted by \"x\"s. The maximum and minimum values \nof V during stable oscillations are denoted by the filled circles. In B, the stable and \nunstable equilibrium points are denoted by the solid and dashed curve, respectively, and \nthe minimum and maximum values of the stable and unstable oscillations are denoted by \nthe filled and unfilled circles, respectively. In B, limit-cycle oscillations appear and \ndisappear via sub-critical Andronov-Hopf bifurcations. The bifurcation diagram (B) was \ncomputed with the LOCBIF program [14]. \n\n\fG. N. Patel, G. S. Cymbalyuk, R. L. Calabrese and S. P. De Weerth \n\n736 \n\nA \n\nExperimental data \n\n100~--~--~----~~ \n\nModeling data \n\n80 \u2022 \n\u2022 \n\n-\nN \n~ \n>. 60 \ng \n~ 40 \nt:T \n~ \nLL 20 \n\n\u2022 \n\u2022 .~~.. \n\u2022 \u2022\u2022\u2022 \n... \n\n\u2022 \n.., \n\n. \n\nB \n\n100 \n\n- 80 \nN ::c ->. 60 \n\n0 c \nQ) \n:l 40 \nt:T \nQ) ... \nLL 20 \n\n\u2022 \n\u2022 \n\n\u2022 \n\n. \n\n\u2022 \u2022\u2022\u2022 \n\n\u2022 \n\u2022 \n\u2022 \n\u2022 \u2022\u2022 \n~ \n\n~. \n\no~--~----~------~ \n\no \n\n20 \n\n10 \nlext (nAmps) \n\n30 \n\n0 \n\n0 \n\n20 \n\n10 \nlext (nAmps) \n\n30 \n\nFigure 4: Frequency dependence of the silicon neuron (A) and the mathematical model \n(B) on the externally applied current. \n\nFor moderate values of lext ([1 nA,34 nA)), the stable and unstable equilibrium points are \nclose to (V, W) ::::: (V H' V L) (Figure 3). In experiments in which lext was varied, we \nobserved a hard loss of the stability of the steady-state equilibrium and a transition into \noscillations at lext = 7.2 nA (I ext = 27.5 nA). In the mathematical model, at the criti(cid:173)\ncal value of lext = 7.7 nA (lext = 27.8 nA), an unstable limit cycle appears via a sub(cid:173)\ncritical Andronov-Hopf bifurcation. This unstable limit cycle merges with the stable limit \ncycle at the fold bifurcation at lext = 3.4 nA (lext = 32.1 nA). Similarly, in the experi(cid:173)\nlext = 2.0 nA \nments, we observed hard \n(I ext = 32.8 nA). Thus, the system demonstrates hysteresis. For example. when \nlext = 20 nA the silicon neuron has only one stable regime, namely, stable oscillations. \nThen if external current is slowly increased to lext = 32.8 nA. the form of oscillations \nchanges. At this critical value of the current, the oscillations suddenly lose stability, and \nonly steady-state equilibrium is stable. Now, when the external current is reduced, the \nsteady-state equilibrium is observed at the values of the current where oscillations were \npreviously exhibited. Thus, within the ranges of externally applied currents (2.0,7.2) and \n(27.5,32.8), oscillations and a steady-state equilibrium are stable regimes as shown in \nFigure 2. \n\nloss of stability of oscillations at \n\n4 Discussion \n\nWe have developed a two-state silicon neuron and a mathematical model that describes the \nbehavior of this neuron. We have shown experimentally and verified mathematically that \nthis silicon neuron has three regions of operation under the variation of its external current \n(one of its parameters). We also perform bifurcation analysis of the mathematical model \nby varying the externally applied current and show that the behaviors exhibited by the sili(cid:173)\ncon neuron under corresponding conditions are in good agreement to those predicted by \nthe bifurcation analysis. \n\nThis analysis and comparison to experiment is an important step toward our understand(cid:173)\ning of a variety of oscillatory hardware networks that we and others are developing. The \n\n\fBifurcation Analysis of a Silicon Neuron \n\n737 \n\nmodel facilitates an understanding of the neurons that the hardware alone does not pro(cid:173)\nvide. In particular for this neuron, the model allows us to determine the location of the \nunstable fixed points and the types of bifurcations that are exhibited. In higher-order sys(cid:173)\ntems, we expect that the model will provide us insight about observed behaviors and \ncomplex bifurcations in the phase space. The good matching between the model and the \nexperimental data described in this paper gives us some confidence that future analysis \nefforts will prove fruitful. \n\nAcknowledgments \n\nS. DeWeerth and G. Patel are funded by NSF grant IBN-95 II 721 , G.S. Cymbalyuk is \nsupported by Russian Foundation of Fundamental Research grant 99-04-49112, R.L. Cal(cid:173)\nabrese and G.S. Cymbalyuk are supported by NIH grants NS24072 and NS34975. \n\nReferences \n\n[1] Van Der Pol, B (1939) Biological rhythms considered as relaxation oscillations In H. \nBremmer and c.J. Bouwkamp (eds) Selected Scientific Papers, Vol 2, North Holland \nPub. Co., 1960. \n\n[2] Mead, C.A. Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA, 1989. \n[3] Simoni, M.E, Patel, G.N., DeWeerth, S.P., & Calabrese, RL. Analog VLSI model of \nthe leech heartbeat elemental oscillator. 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I. , Kuznetsov, Yu.A., Levitin, v.v., Nikolaev, E.V. (1993) Continuation \ntechniques and interactive software for bifurcation analysis of ODEs and iterated \nmaps. Physica D 62 (1-4): 360-367. \n\n\f", "award": [], "sourceid": 1690, "authors": [{"given_name": "Girish", "family_name": "Patel", "institution": null}, {"given_name": "Gennady", "family_name": "Cymbalyuk", "institution": null}, {"given_name": "Ronald", "family_name": "Calabrese", "institution": null}, {"given_name": "Stephen", "family_name": "DeWeerth", "institution": null}]}