{"title": "Optoelectronic Implementation of a FitzHugh-Nagumo Neural Model", "book": "Advances in Neural Information Processing Systems", "page_first": 1091, "page_last": 1098, "abstract": null, "full_text": "Optoelectronic Implementation of a\nFitzHugh-Nagumo Neural Model\n\nAlexandre R.S. Romariz , Kelvin Wagner\nOptoelectronic Computing Systems Center\n\nUniversity of Colorado, Boulder, CO, USA 80309-0425\n\nromariz@colorado.edu\n\nAbstract\n\nAn optoelectronic implementation of a spiking neuron model based on\nthe FitzHugh-Nagumo equations is presented. A tunable semiconduc-\ntor laser source and a spectral \ufb01lter provide a nonlinear mapping from\ndriver voltage to detected signal. Linear electronic feedback completes\nthe implementation, which allows either electronic or optical input sig-\nnals. Experimental results for a single system and numeric results of\nmodel interaction con\ufb01rm that important features of spiking neural mod-\nels can be implemented through this approach.\n\n1 Introduction\n\nBiologically-inspired computation paradigms take different levels of abstraction when\nmodeling neural dynamics. The production of action potentials or spikes has been ab-\nstracted away in many rate-based neurodynamic models, but recently this feature has gained\nrenewed interest [1, 2]. A computational paradigm that takes into account the timing of\nspikes (instead of spike rates only) might be more ef\ufb01cient for signal representation and\nprocessing, especially at short time windows [3, 4, 5].\n\nOptics technology provides high bandwidth and massive parallelism for information pro-\ncessing. However, the implementation of digital primitives have not as yet proved compet-\nitive against the scalability and low power operation of digital electronic gates. It is then\nnatural to explore the features of optics for different computational paradigms. Arti\ufb01cial\nneural networks promise an excellent match to the capabilities of optics, as they emphasize\nsimple analog operations, parallelism and adaptive interconnection[6, 7, 8, 9].\n\nOptical implementations of Arti\ufb01cial Neural Networks have to deal with the problem of\nrepresenting the nonlinear activation functions that de\ufb01ne the input-output mappings for\neach neuron. Although nonlinear optics has been suggested for implementing neurons,\nhybrid optoelectronic systems, where the task of producing nonlinearity is given to the\nelectronic circuits, may be more practical [10, 11]. In the case of pulsing neurons, the task\nseems more dif\ufb01cult still, for instead of a nonlinear static map we are required to imple-\nment a nonlinear dynamical system. Several possibilities for the implementation of pulsed\noptical neurons can be considered, including smart pixel pulsed electronic circuits with op-\n\n\u0001 On leave from the Electrical Engineering Department, University of Bras\u00b4\u0131lia, Brazil\n\n\ftical inputs [12], pulsing laser cavity feedback dynamics [13] and competitive-cooperative\nphosphor feedback [14].\n\nIn this paper we demonstrate and evaluate an optoelectronic implementation of an arti\ufb01cial\nspiking neuron, based on the FitzHugh-Nagumo equations. The proposed implementation\nuses wavelength tunability of a laser source and a birefringent crystal to produce a nonlinear\nmapping from driving voltage to detected optical output [15]. Linear electronic feedback to\nthe laser drive current completes the physical implementation of this model neuron. Inputs\ncan be presented optically or electronically, and output signals are also readily available as\noptical or electronic pulses.\n\nThis work is organized as follows. Section 2 reviews the FitzHugh-Nagumo equations and\ndescribes the particular optoelectronic spiking neuron implementation we propose here. In\nSection 3 we analyze and illustrate dynamical properties of the model. Experimental results\nof the optoelectronic system implementing one model are presented in Section 4. Numeric\nresults that illustrate features of the interaction between models are shown in Section 5.\n\n2 Modi\ufb01ed FN Neural Model and optoelectronic implementation\n\nThe FitzHugh-Nagumo neuron model [16, 17] is appealing for physical implementation, as\nit is fairly simple and completely described by a pair of coupled differential equations:\n\n(1)\n\nbeen previously implemented in CMOS integrated electronics [18].\n\n\u000e\u0010\u000f\n\u0005\u0007\u0006\t\b\u000b\n\u0012\u0011\u0007\u0013\u0015\u0014\u0016\u0006\t\b\u000b\n\u0018\u0017\u001a\u0019\u001b\u0006\t\b\u000b\n\n !\u0005\u0007\u0006\t\b\u000b\n\u0004\u0013\"\u0014\u0016\u0006#\b\u000b\n$\u0017&%\n\nis an excitable state variable that exhibits bi-stability as a result of the nonlinear\nis a linear recovery variable, bringing the neuron back to a resting state.\n\n\u0005\u0007\u0006\t\b\u000b\n\r\f\n\u0002\u0001\u0004\u0003\n\u0014\u001f\u0006\t\b\u000b\n\r\f\n\u001d\u001c\u001e\u0003\nwhere\u0005\nterm, and\u0014\n\u000e\u0010\u000f\n\u0005'\u0011\n\u0005(\u0011 is a third-degree polynomial[16, 17]. This model has\nIn the original model proposal,\u000e\u0010\u000f\n)*\u0011 by using the nonlinear response of linear optical systems to varia-\nnonlinear function\u000e\u0010\u000f\n\nIn optical implementation of neural networks, the required nonlinear functions are usually\nperformed through electronic devices, with adaptive linear interconnection done in the op-\ntical domain. We here explore the possibility of optical implementation of the required\n\nConsider a birefringent material placed between crossed polarizers. Even though propa-\ngation of the \ufb01eld through the material is a linear phenomenon (a linear phase difference\namong orthogonal polarization components is generated), the output power as a function\nof incident wavelength is sinusoidal, according to\n\ntions of the wavelength.\n\n(2)\n\nis the responsivity (in\nis the optical power incident on the detector, which is a function of the laser\nis the optical path difference (OPD) resulting from propagation through\n\n\u0005'+-,\u0012.\u0010\f0/21 det\f0/43657\u0006#89\n;:=A@CBEDGF\n\u0006#89\nJI\nis the transimpedance gain of the detector ampli\ufb01er, 3\n\u0006#89\n\nwhere/\nA/W),57\u0006#89\n\ndrive current8 ,\nthe birefringent material andH\nticular, an input current8 produces a small modulation in the radiation wavelength H\n\u0006\t8K\n .\nH variation in Equation 2, we \ufb01nd a nonlinear mapping from driving\n\nIn semiconductor lasers, and Vertical Cavity Surface Emitting Lasers (VCSELs) in par-\n\nLinearizing the\nvoltage to detected signal:\n\nis the laser wavelength.\n\n(3)\n\nLJM\n\n+-,\u0012.\n\n\fN\u000e\u0010\u0006#\u0005;\nO\fQP7\u0006\t\u0005R\n\u000b:\u000b~~A@\n\n\u0006#\u0005\u001f\u0013UT\u0007V;\n\nTAW\n\nH\nF\n\u0005\nB\nD\nI\n\f\u2212\n\n+\n\n\u2212\n\n+\n\nv\n\ni\n\nDriver\n\nu\n\nw\n\nf(v)\n\nPD\n\nno\n\nne\nBirefringent Crystal\n\nMirror\n\nVCSEL\n\nCollimation\n\nPBS\n\n(a)\n\nDetected Optical Signal\n\n)\n\nV\n\n(\nt\ne\nD\nV\n\n0.5\n\n0.4\n\n0.3\n\n0.2\n\n0.1\n\n0.0\n\n0.05\n\n0.10\n\n0.15\n0.25\nDriver Voltage (V)\n\n0.20\n\n0.30\n\n0.35\n\n(b)\n\nFigure 1: a Experimental setup for the wavelength-based nonlinear oscillator, with simpli-\n\ufb01ed view of the electronic feedback. b Experimental evidence of nonlinear mapping from\ndriver voltage to detected signal (open loop), as a result of wavelength modulation as well\nas laser threshold and saturation.\n\nprocess, as well as nonlinear phenomena such as laser threshold and saturation.\n\nthrough the driver\nincludes all conversion factors in the detection\n\nwhere\u0005\ntransconductance) and the functionP7\u0006\t\u0005R\n\nis the driving voltage (linearly converted to an input current8\n\nA simple nonlinear feedback loop can now be established, by feeding the detected signal\nback to the driver. This basic arrangement has been used to investigate chaotic behavior\nin delayed-feedback tunable lasers [15] . It is used here as the nonlinearity for an optical\nself-pulsing mechanism in order to implement neural-like pulses based on the following\ndynamical system\n\nAgain\u0005\n\n\u0005\u0007\u0006\t\b\u000b\n\n\u0002\u0001\u0004\u0003\n\u0014\u0016\u0006\t\b\u000b\n\n\u0006#\b\u000b\nO\u0013\u0015\u0014\u0016\u0006\t\b\u000b\n$\u0017\u001a\u0019\u001b\u0006\t\b\u000b\n\n\u0005\u0007\u0006\t\b\u000b\n\u0012\u0011\u0007\u0013\u0015\u0005\n\u000e\u0010\u000f\n !\u0005\u0007\u0006\t\b\u000b\n\u0004\u0013\"\u0014\u0016\u0006#\b\u000b\n\u001b\u0017\u001a%\na relatively slow recovery variable, so thatJ\u0001\n\nis a fast state variable, and\u0014\n\n.\nThe experimental setup is shown in Figure 1a. Light from the tunable source is collimated\nand propagates through a piece of birefringent crystal. The crystal fast and slow axis are\nat 45 degrees to the polarizer and analyzer passing axis. The effective propagation length\nthrough the crystal (and corresponding wavelength selectivity) is doubled with the use of a\nmirror. A polarizing beam splitter acts as both polarizer and analyzer. A simpli\ufb01ed view\nof the electronic feedback is also shown. Leaky integrators and linear analog summations\nimplement the linear part of Equation 4, while the nonlinear response (in intensity) of the\n\n(4)\n\n\u001d\u001c\n\noptical \ufb01lter implements\u000e\u0010\u0006#\u0005;\n .\n\nA VCSEL was used as tunable laser source. These vertical-cavity semiconductor lasers\nhave, when compared to edge-emitting diode lasers, larger separation between longitudinal\nmodes, more circularly-symmetric beams and lower fabrication costs [19]. As the input\ncurrent is increased, the heating of the cavity red-shifts the resonant wavelength [20], and\nthis is the main mechanism we are exploring for wavelength modulation.\n\nAn experimental veri\ufb01cation of the expected sinusoidal variation of detected power with\nmodulation voltage is given in Figure 1b. A slow (800Hz) modulation ramp was applied to\nthe driver, and the detected power variation was acquired. From this information, the static\ntransfer function shown in the right part of the \ufb01gure was calculated. Unlike the experiment\nwith a DBR laser diode reported by Goedgebuer et al.\n[15], it is apparent that current\nmodulation is affecting not only wavelength (and hence effective optical path difference\namong polarization components) but overall output power as well. Modulation depth is\nlimited (non-zero troughs in the sinusoidal variation), which we attribute to the multiple\ntransverse modes that the device supports. However, as we are going to be operating near\n\n\f\n\n\u001c\n\u0003\n\f\n\u0001\n\fFigure 2: Continuous line: trajectory of the system under strong input, obtained by nu-\nmeric integration (\n\n.\u0002\u0001 -order Runge-Kutta) of Equation 4. Arrows represent the strength of\nthe derivatives at a particular point in state space. Dashed line: nullcline \u0003\n\u0005\u001a\f\u0004\u0003 . Dash-\n\f\u0005\u0003 . Stability analysis show that the equilibrium point where the\ndotted line: nullcline \u0003\nnullclines meet is unstable, so the limit cycle is the sole attractor. Parameters\n\u001d\u0001 ,\nV,T\n,\u0019\n\f\n\u0003\n 0\f\nL\u0007\u0006\t\b\n\nthe \ufb01rst maximum (see Section 3), the power variation over successive maxima should not\naffect the dynamical properties of the closed-loop system. The relatively smooth curve\nobtained indicates that no mode hops occurred for this driving current range, which was\nindeed con\ufb01rmed with Optical Spectrum Analyzer measurements.\n\n\u0003 ,%\n\n\u0006\t\b\u0007\b\n\n\u000b V.\n\n\f\n\u0003\n\n\u0003\u0007\u000b\r\f V,T\u0007VC\f\u000e\u0003\n\n\u0006?L\u000f\b\n\n3 Simulations\n\nFitzHugh-Nagumo models are known to have so-called class II neural excitability (see [21]\nfor a review). This class is characterized by an Andronov-Hopf bifurcation for increasing\nexcitation, and exhibits some dynamical phenomena that are not present in integrate-and-\n\ufb01re dynamics. For equal intensity input pulses, integrators will respond maximally to the\npulse train with lowest inter-spike interval. Class II neurons have resonant response to a\nrange of input frequencies. There are non-trivial forms of excitation in resonator mod-\nels that are not matched by integrators: the former can produce a spike at the end of an\ninhibitory pulse, and conversely, can have a limit cycle condition interrupted (with the sys-\ntem recovering to rest) by an excitatory pulse.\n\nWe have veri\ufb01ed that these characteristics are maintained in the modi\ufb01ed optical model,\ndespite the use of a sinusoidal nonlinearity instead of the original \u000b\u0011\u0010\nfunction. Stability analysis based on the Jacobian of the dynamical system (Equation 4)\nshows an Andronov-Hopf bifurcation, as in the original model. Limit cycle interruption\nthrough exciting pulses is shown in Section 5.\n\n+ degree polynomial\n\nFigure 2 shows a typical limit-cycle trajectory, for parameter values that match conditions\nof the experiment reported in Section 4. Parameters were chosen so that a typical excursion\nin modulation voltage goes from the dead zone (below the lasing threshold) to around the\n\ufb01rst peak in the nonlinear detector transfer function. This is an interesting choice because\nthe optical output is only present during spiking, and can be used directly as an input to\nother optoelectronic neurons.\n\n\n\u0014\n\u001c\n\f\nL\n\u0003\nL\n\f\n\u0013\nW\n\u0006\n\f0.180\n0.155\n0.130\n0.105\n0.080\n\n10\n\n0.12\n0.09\n0.06\n0.03\n0.00\n\n10\n\n0.100\n0.075\n0.050\n0.025\n0.000\n\n10\n\nDriver Voltage(V)\n\n20\n\n20\n\n30\nTime(t\nv)\nRecovery\n\n40\n\n50\n\n30\n\nDetectd signal\n\n40\n\n50\n\n20\n\n30\n\n40\n\n50\n\n(a)\n\n(b)\n\nFigure 3: Dynamical system response to strong constant input. a Simulation results. Pa-\n\nms.\n\nrameters as in Figure 2. b Experimental results. Parameters:\n\nInput (V)\n\nms,\n\n0.2\n0.1\n0.0\n0\n\n0.2\n0.0\n-0.2\n\n0.10\n\n0.05\n\n0.00\n\n0.10\n\n0.05\n\n0.00\n\n0\n\n0\n\n0\n\n20\n\n20\n\n20\n\n20\n\n60\n\n80\n\n100\n\n40\n\nTime(t\n\nv)\n\nDriver Voltage (V)\n\n40\n60\nRecovery (V)\n\n40\n\n60\n\nDetected Optical Signal (V)\n\n40\n\n60\n\na\n\n80\n\n80\n\n80\n\n100\n\n100\n\n100\n\n\f\u000e\u0003\n\n\u0006SL\n\nb\n\nFigure 4: (a): Simulated response to a train of pulses. Parameters as in Figure 2.\nExperimental Results. Parameters as in Figure 3.\n\n(b):\n\n4 Experimental Results\n\nFigure 3 presents a comparison between simulated waveforms for the various dynamic\nvariables involved (as the system performs the trajectory depicted in Figure 2) and the\nexperimental results obtained with the system described in Figure 1, revealing a good\nagreement between simulated and experimental waveforms. The double-peak in the op-\ntical variable can be understood by following the trajectory indicated in Figure 2, bearing\nin mind the non-monotonic mapping from driver voltage to detected signal. The decrease in\nincreases produces initially an increase\n\nin detected power, and thus the second, broader peak at the end of the cycle.\n\ndriver voltage observed as the recovery variable\u0014\n\nThe production of sustained oscillations for constant input is one of the desired characteris-\ntics of the model, but in a network, neurons will mostly communicate through their pulsed\noutput. The response of the system to pulsed inputs can be seen in Figure 4. The output\noptical signal response is all-or-none, but sub-threshold integration of weak inputs is being\nslowly returns\nto 0, a new excitation just after a pulse is less likely, which can be seen at the response to\nthe third pulse. The experimentally observed waveforms agree with the simulations, though\ndetails of the pulsing in the optical output are different.\n\nperformed, as the waveform for driver voltage shows in the \ufb01rst pulse. As\u0014\n\n\u0001\n\u001c\n\f\nL\n\fBias\n\no\n\n2\n\n0\n\ni\n\na\n\nd\na\nr\n\n4\n\n2\n\n0\n\n-2\n\n-4\n0\n\nPulse advance vs input pulse phase\n\nNo spikes\n\n2\n\n4\n\nInput Pulse Phase (rad)\n\n6\n\nb\n\nFigure 5: Numeric illustration of the effect of input timing on the advance of the next spike,\nin the modi\ufb01ed FitzHugh-Nagumo system. a: Schematic view of simulation. See text for\ndetails. b: Phase advance as a function of input phase. Bias 0.103V. Input pulse height\n10 mV, duration\n\n\u0001 . Dynamic system parameters as in Figure 2.\n\n5 Coupling\n\nOne of the main motivations for using optical technology in neural network implementation\nis the possibility of massive interconnection, and so the de\ufb01nition of coupling techniques,\nand the study of adaptation algorithms compatible with the dynamical properties of the\nexperimentally-demonstrated oscillators are the current focus of this research.\n\nThe most elegant optical implementation of adaptive interconnection is through dynamic\nvolume holography[6, 11], but that requires a set of coherent optical signals, not what we\nhave with an array of pulse emitters. In contrast, the matrix-vector multiplier architecture\nallows parallel interconnection of incoherent optical signals, and has been used to demon-\nstrate implementations of the Hop\ufb01eld model [7] and Boltzman machines [9].\n\nAn interesting aspect of the coupled dynamics in oscillators exhibiting class II excitability\nis that the timing of an input pulse can result in advance or retardation of the next spike [22].\nThis is potentially relevant for hardware implementation, as the excitatory (i.e., inducing an\nearly spike) or inhibitory character of the connection might be controlled without changing\nsigns of the coupling strength.\n\nIn Figure 5 we show a simulation illustrating the effect of input pulse timing in advancing\nthe output spike. A constant input to a model neuron (Equation 4) was maintained, pro-\nducing periodic spiking. A second, positive, pulsed input was activated in between spikes,\nand the effect of this coupling on the advance or retardation of the next spike was veri\ufb01ed\n\u0003 ) with\nas the timing of the input was varied. A region of output spike retardation (\n\u0002\u0001\u0004\u0003\u0006\u0005\nexcitatory pulsed input can be seen. Even more interesting, for phases around\nrad relative\nto the latest spike, the excitatory pulse can terminate periodic spiking altogether.\n\nThis phenomenon is seen in detail in Figure 6, where both the time waveforms and state-\nspace trajectories are shown. For this particular condition, the equilibrium point of the\nsystem is stable. When correctly timed, the short excitatory pulse forces the system out\nof its limit cycle, into the basin of attraction of the stable equilibrium, hence stopping the\nperiodic spiking. As the individual models used in this simulations were shown to match\nexperimental implementations in Section 4, we expect to observe the same kind of effect\nin the coupling of the optoelectronic oscillators.\n\nD\nf\np\nf\nL\n\u0003\n\nD\n\fV\n\n0.016\n0.008\n0.000\n\nV\n\n0.2\n0.1\n0.0\n\nV\n\n0.10\n0.05\n0.00\n\n40\n\n60\n\n40\n\n60\n\n40\n\n60\n\nInput\n\nState Space Trajectory\n\n80\n\n100\n\nTime(t.c.)\n\nDriving Voltage\n\n80\n\n100\n\nTime(t.c.)\n\nOutput \n\n80\n\n100\n\nTime(t.c.)\n\na\n\n120\n\n140\n\n120\n\n140\n\n120\n\n140\n\n 0.0800\n\n 0.0675\n\n 0.0550\n\n 0.0425\n\nw\n\n 0.0300\n\n 0.0175\n\n 0.0050\n\n-0.0075\n\n-0.0200\n\n 0.1400\n\n 0.1600\n\n 0.0800\n\n 0.1000\n\n 0.1200\n\nv\n\nb\n\n. Other parameters as in Figure 2. (b): Same results in state space. Continuous line:\n\nUnperturbed trajectory. Dotted Line: Trajectory during excitatory pulse.\n\nFigure 6: (a): Simulated response illustrating return to stability with excitatory pulse.\u0019\n\u0006SL'L\u0001\n\n6 Ongoing work and conclusions\n\nImplementation of a modi\ufb01ed FN neuron model with a nonlinear transfer function realized\nwith a wavelength-tuned VCSEL source, a linear optical spectral \ufb01lter and linear elec-\ntronic feedback was demonstrated. The system dynamical behavior agrees with simulated\nresponses, and exhibits some of the basic features of neuron dynamics that are currently\nbeing investigated in the area of spiking neural networks.\n\nFurther experiments are being done to demonstrate coupling effects like the ones described\nin Section 5. In particular, the use of external optical signals directly onto the detector\nto implement optical coupling has been demonstrated. Feedback circuit simpli\ufb01cation is\nanother important aspect, since we are interested in implementing large arrays of spiking\nneurons. With enough detection gain, Equation 4 should be implementable with simple\nRLC circuits, as in the original work by Nagumo[17].\n\n\u0002\u0004\u0003\n\nResults reported here were obtained at low frequency (1-100 KHz), limited by ampli\ufb01er\nand detector bandwidths. With faster electronics and detectors, the limiting factor in this\narrangement would be the time constant for thermal expansion of the VCSEL cavity, which\n. Pulsing operation at 1.2 MHz has been obtained in our latest experiments.\nis around 1\n\nEven faster operation is possible when using the internal dynamics of wavelength modu-\nlation itself, instead of external electronic feedback. In addition to the thermally-induced\nmodulation of wavelength, carrier injection modi\ufb01es the index of refraction of the active re-\ngion directly, which results in an opposite wavelength shift. By using this carrier injection\neffect to implement the recovery variable, feedback electronics is simpli\ufb01ed and a much\nfaster time constant controls the model dynamics. Optical coupling of VCSELs has the\npotential to generate over 40GHz pulsations [23]. Our goal is to investigate those optical\noscillators as a technology for implementing fast networks of spiking arti\ufb01cial neurons.\n\nAcknowledgments\n\nThis research is supported in part by a Doctorate Scholarship to the \ufb01rst author from the\nBrazilian Council for Scienti\ufb01c and Technological Development, CNPq.\n\n\f\n\u0003\n\fReferences\n[1] F. Rieke, D. Warland, R.R. von Steveninck, and W. Bialek. Spikes: Exploring the Neural Code.\n\nMIT Press, Cambridge, USA, 1997.\n\n[2] T.J. Sejnowski. Neural pulse coding. 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