An excitable electronic circuit as a sensory neuron model
Bruno N. S. Medeiros, Victor Minces, Gabriel B. Mindlin, Mauro Copelli, José R. Rios Leite
aa r X i v : . [ phy s i c s . b i o - ph ] O c t October 24, 2018 3:50 medeiros-ijbc-2011
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
An excitable electronic circuit as a sensory neuron model
Bruno N. S. Medeiros , Victor Minces , Gabriel B. Mindlin , Mauro Copelli , Jos´e R. Rios Leite Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife, PE, Brazil. Department of Cognitive Neuroscience, University of California San Diego, 92093-0515, La Jolla, CA,USA Departamento de F´ısica, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, 1428,Buenos Aires, Argentina
Received (to be inserted by publisher)
An electronic circuit device, inspired on the FitzHugh-Nagumo model of neuronal excitability,was constructed and shown to operate with characteristics compatible with those of biologicalsensory neurons. The nonlinear dynamical model of the electronics quantitatively reproduces theexperimental observations on the circuit, including the Hopf bifurcation at the onset of tonicspiking. Moreover, we have implemented an analog noise generator as a source to study thevariability of the spike trains. When the circuit is in the excitable regime, coherence resonance isobserved. At sufficiently low noise intensity the spike trains have Poisson statistics, as in manybiological neurons. The transfer function of the stochastic spike trains has a dynamic range of6 dB, close to experimental values for real olfactory receptor neurons.
Keywords : Electronic circuit, Excitable element, Coherence resonance, Dynamic range.
1. Introduction
Ever since the pioneering work of Hodgkin & Huxley [1952], the biophysical mechanisms underlying thegeneration and propagation of action potentials (spikes) in neurons have been described with increasingdetail, ranging from the discovery of new types of ion channels to the study of intracellular calciumdynamics [Hille, 2001]. No matter how interesting, these new findings have helped little in our understandingof collective neuronal phenomena, which remain a daunting task in face of the interplay among high-dimensionality, noise and nonlinearity (see e.g. Chialvo [2010] for a recent review). The challenge shouldnonetheless be faced: the solution of issues at the frontiers of current-day neuroscience, like e.g. grandmothercell [Barlow, 1972] versus population coding [Young & Yamane, 1992], or firing rate versus spike-timecoding [Rieke et al. , 1999] will likely be grounded on our success in this endeavor.In fact, theoretical progress in this front has been achieved in recent years with very simple models.One such example is the proposed solution for the century-old problem of the origin of psychophysicalresponse curves [Copelli et al. , 2002; Kinouchi & Copelli, 2006]. Steven’s psychophysical law states thatthe psychological perception F of a physical stimulus (e.g. light, or odorant) of intensity h is a power law F ∝ h s , with experimental values of the Stevens exponent s fluctuating around s ≃ .
5. Compared toa linear response, psychophysical nonlinear responses have at least one evolutionarily favorable property: ctober 24, 2018 3:50 medeiros-ijbc-2011 Medeiros et al. V b V a V in V out V + R =10kR =1M Ω R =10k Ω R =1k Ω R =10k Ω V_ TL071C = 1nF or 50pF Ω Fig. 1. Excitable electronic circuit. V a and V b = − V a are the operational amplifier supply voltages. V in is an input voltage,corresponding to an external stimulus. We describe the circuit as a two-dimensional dynamical system on the variables V − and V out (see Eqs. (2)). they amplify weaker stimulus, i. e. they have a larger dynamic range. But how do the Stevens exponentsarise in the nervous system?At first, this question seems puzzling because single neurons typically have small dynamicranges [Rospars et al. , 2000]. A theoretical solution recently proposed involves a collective phenomenon:excitable waves are generated by the incoming stimuli and propagate “laterally” among excitable neu-rons, thereby amplifying the system response (in comparison to what would be observed in the absenceof the coupling). Interestingly, this amplification mechanism is self-limited: under intense stimulation, forinstance, a large number of excitable waves can be created, but owing to refractoriness they annihilateupon collision. The enhancement of dynamic range in this model is therefore governed by the low-stimulusamplification [Copelli et al. , 2002; Kinouchi & Copelli, 2006]. Robustness of these results has been tested atdifferent modeling levels [Copelli et al. , 2005; Ribeiro & Copelli, 2008; Assis & Copelli, 2008; Publio et al. ,2009], showing that the degree of biophysical realism in the model of each neuron is less relevant to theglobal dynamics than the topology of the network [Copelli & Kinouchi, 2005; Copelli & Campos, 2007;Ribeiro & Copelli, 2008; Assis & Copelli, 2008; Gollo et al. , 2009]. This phenomenon has also been studiedanalytically [Furtado & Copelli, 2006; Larremore et al. , 2011] and was recently confirmed experimentallyin cortical slices [Shew et al. , 2009].The appeal of a sensory system with large dynamic range based on a network of simple excitableunits, each with small dynamic range, goes beyond basic research in neuroscience. The idea could bereversed, leading to biologically inspired artificial sensors, which have been used in a variety of scenarios (seee.g. de Souza et al. [1999]).There are several electronic circuits reported in the literature which have been designed to presenta neuron-like dynamical response. The rationale behind those efforts was to dynamically interact withbiological neurons rather than stimulating them using response independent current commands. In this way,electronic circuits which analogically integrated the Hindmarsh and Rose equations [Szucs et al. , 2000] werecoupled to the neurons of a preparation of lobster pyloric CPG neurons. This allowed to show that regularitycould emerge as a collective dynamical property of units which individually presented complex dynamics.In another set of experiments, electronic neurons interacting with a biological preparation were used tounveil which dynamical properties of a neural network depend on the bifurcation leading to excitationfor the units, rather than on the details of the neural dynamics. To carry out this program, a standardform for class I excitable dynamics was analogically integrated with a circuit, which was used to replacea neuron in a midbody ganglion of the leech Hirudo medicinalis [Aliaga et al. , 2003]. The responses underthe stimulation of both the natural preparation and the one with a replaced neuron were found to besimilar. Beyond the possibility of interacting with neurons through a dynamically sensible way, these effortsprovide empirical support to the program of studying neural processes through simple and relatively lowdimensional dynamical systems. Depending on the question under study, it might be desirable to be able toctober 24, 2018 3:50 medeiros-ijbc-2011 An excitable electronic circuit as a sensory neuron model establish a closer link between the device and a neuron. In this spirit, a device implementing a conductancemodel was recently proposed [Sitt & Aliaga, 2007].These circuits, however, have two limitations for our purposes. First, they are still too complex to bereplicated in large scale. Second, they do not have a controllable noise source to produce stochastic spiketrains, a feature that is common to the both models [Copelli et al. , 2002] and real neurons [Dayan & Abott,2001; Mainen & Sejnowski, 1995; Petracchi et al. , 1995]. The present work is a first step in this direction.We propose an excitable electronic circuit which can serve as a building block of an electronic sensor. Theadvantages of its extreme simplicity are twofold: it allows for scalability and, at the same time, simplemathematical modeling.The paper is organized as follows. In section 2, we describe the electronic circuit and the equationsthat model its dynamics. In section 3, we introduce noise from a simple analog noise generator at the inputof the excitable circuit and study the statistical properties of the resulting spike trains and show that itcan exhibit Poisson statistics as well as coherence resonance, as expected. In section 4 we evaluate thedynamic range of the excitable circuit and show that it is comparable to that of single sensor neurons.
2. Dynamic model
The circuit we propose is shown in Fig. 1. It is composed of five resistors, one capacitor and one operationalamplifier. The voltage V in corresponds to an external stimulus, which can be e.g. a constant or the sum ofDC and noise voltages. In our electronic neuron, the operational amplifier behaves as a simple comparatorcircuit, for which we use the following nonlinear model: dV out dt = S sign [ V b − V out + ( V a − V b )Θ( V + − V − )] , (1)where Θ is the Heaviside function and S is the op-amp slew rate (whose datasheet value for the simpleTL071 in the circuit is S = 16 V/s). As usual, symmetric supply voltages V b = − V a were used.Assuming R ≫ R , R and applying Kirchhoff’s laws, we arrive at a two-dimensional dynamic modelon the variables V out and V − : dV out dt = V c ε sign [ V b − V out + ( V a − V b )Θ( αV out − V − )] , (2a) dV − dt = 1 R C (cid:2) βV out + γV in − V − (cid:3) . (2b)where α ≡ R / ( R + R ), β ≡ R / ( R + R ) and γ ≡ R / ( R + R ). V c = 10 V is a characteristic voltageof the same order of magnitude of the supply voltages, and we have defined ε ≡ V c /S as a characteristic(short) time scale. To avoid the possibility that the system (2) has more than one fixed point, we require β > α . In terms of the variables v ≡ V out V c , (3a) w ≡ V − V c , (3b)the equations can be rewritten in dimensionless form˙ v = sign (cid:18) b − v + ( a − b )1 + e − ( αv − w ) /x (cid:19) , (4a)˙ w = φ [ βv + γj − w ] , (4b)where we defined the dimensionless groups: τ ≡ tε ; a = V a V c ; b = V b V c ; φ = εR C ; j = V in V c , (5)ctober 24, 2018 3:50 medeiros-ijbc-2011 Medeiros et al. -0.1-0.05 0 0.05 0.1 -1 -0.5 0 0.5 1 w va) -0.1-0.05 0 0.05 0.1 -1 -0.5 0 0.5 1 w va) -1.5-1-0.5 0 0.5 1 1.5-15-10 -5 0 5 10 15 V - ( V ) V out (V)b) 0 0.5 1 1.5-10 -5 0 5 10 f ( k H z ) V in (V)c)-15-10-5 0 5 10 15 0 1 2 3 4 V ou t ( V ) V - ( V ) time (ms)d) -10-5 0 5 10 0 1 2 3 4 V m ( V ) time (ms)e) Fig. 2. a) Nullclines of system (4) for a = 1, b = 1, α = 0 . β = 0 . γ = 0 . j = 0, φ = 0 .
01 and x = 9 × − :solid black line for the ˙ v = 0 nullcline and dashed black line for the ˙ w = 0 nullcline. The fixed point is unstable and thetrajectories are attracted to a limit cycle (red solid line). b) Experimental limit cycle (black dots) and numerical integrationof the model (red solid line) for x = 1 × − , V a = 10 V, V b = −
10 V, V in = − φ = 5 × − (other parameters arethe same as in (a)). c) Experimental frequency response f to the external DC stimulus V in (black dots) and the same for thenumerical integration of the model (red line). d) Comparison between experimental time series of V out and V − (black circlesand triangles, respectively) with numerical integration of the model (red and green lines, respectively). e) Experimental (blackdots) and numerical (red line) spike trains obtained from the analog subtraction V m of the dynamical variables (see text fordetails). and replaced Θ by the continuous function˜Θ( x ; x ) = 11 + e − x/x (6)for the purpose of numerical integration and derivation (see below). Note that ˜Θ → Θ as x →
0. Theconstant φ ≪ R C ultimately controls the overall time scale of the problem.As shown in Fig. 2a (black lines), the nullclines ˙ v = 0 and ˙ w = 0 of Eqs. (4) resemble those of theFitzHugh-Nagumo model for neuronal excitability, with one fast ( v or V out ) and one slow ( w or V − ) variable.In the limit x →
0, the cubic-like ˙ v = 0 nullcline becomes piecewise linear. When the fixed point sits at itsouter branches, it is stable. It loses stability in a Hopf bifurcation as the w nullcline crosses the v nullclineat its central branch, so trajectories are attracted to a limit cycle (red line) with nonzero frequency f (i.e. f changes discontinuously at the bifurcation). Below the Hopf bifurcation, the circuit is said to be type-IIexcitable [Rinzel & Ermentrout, 1998].There is good quantitative agreement between experimental data from the circuit and the numericalintegration, as can be seen in Fig. 2b, c, d and e. Note that through an analog subtraction V m ≡ . V − − . V out (see also Fig. 3b) the circuit exhibits the spikes typical of neuronal membrane potentials (Fig. 2e).We emphasize that in Fig. 2 experimental and numerical data agree without any fitting parameter, as longas x is sufficiently small ( . − ).
3. Noise addition and coherence resonance
So far we have discussed the response of the excitable circuit under DC stimulation. Biological neurons,however, can show highly variable responses, even when subjected to a presumably constant stimulus.ctober 24, 2018 3:50 medeiros-ijbc-2011
An excitable electronic circuit as a sensory neuron model R A3 R A4 V noise V DC V in V out V _ V _ AnalogadditionV noise
Excitablecircuit Analogsubtraction V _ V out m b) 1 2 43Buffer R A1 R A2 a) Fig. 3. a) An analog noise generator based on the amplification of transistors thermal noise. Noise amplification is givenby A = [( R A + R A ) /R A ]( R A /R A ). b) Block diagram of the circuit used to verify the excitability of the circuit pre-sented in Fig. 1. Analog addition and subtraction (see text for details) are performed with standard TL074 op-amp opera-tions [Senturia & Wedlock, 1981]. -8-4 0 4 8 V m ( V ) experimentalb) -8-4 0 4 8 0 0.1 0.2 0.3 0.4 0.5 V m ( V ) time (s) numericalc) t p -0.1 0 0.1 -2 -1 0 1 2 w v a) -0.1 0 0.1 -2 -1 0 1 2 w v a) R p noise amplification A d) Poisson
Fig. 4. a) Numerical phase plane trajectory (red line) due to noise excitation. Without noise the system would stay in aresting state at the fixed point (white dot). Experimental (b) and numerical (c) spike train series are shown when the systemis in the excitable state (stable fixed point as shown in (a)). d) Experimental coherence resonance curve for C = 50 pF, V a = 12 V = − V b and V DC = − .
826 V (see Fig. 1). Each point corresponds to an average over 10 s time series.
Examples range from highly variable responses olfactory receptor neurons (ORNs) to presentation of iden-tical puffs of odorants [Rospars et al. , 2000], to cortical cells stimulated with a constant current via anintracellular electrode [Mainen & Sejnowski, 1995]. In an attempt to endow our excitable circuits with thevariability in the spike trains observed in biological neurons, we propose the simple analog noise generatorshown in Fig. 3a. Once more, its simplicity allows one to attach independent noise generators to eachexcitable circuit when connecting them in a network.The circuit in Fig. 3a provides a two-stage amplification control via two operational amplifiers to thethermal noise produced by the KN2222 transistors. Its output voltage V noise is approximately a Gaussianwhite noise voltage with a cutoff frequency around 1 kHz.To obtain variable spike trains, the stimulus V in consists in the analog addition of V DC and V noise (seectober 24, 2018 3:50 medeiros-ijbc-2011 Medeiros et al. blocks 1 and 2 in Fig. 3b). In the model, this corresponds to replacing Eq. (4b) with˙ w = φ [ βv + γj + Dξ ( t ) − w ] , (7)where D grows linearly with the gain in the noise amplification A (which in turn is controlled by thevariable resistors shown in Fig. 3).Setting V DC below the Hopf bifurcation, the circuit sits at a stable fixed point at the right branch ofthe ˙ v = 0 nullcline, from which it eventually departs owing to noise (Fig. 4a). This generates spike trainswith variable interspike intervals t p , as shown in Fig. 4b and c.We now show that the interplay between noise and excitability behaves as expected in our simplecircuits. Pikovsky & Kurths [1997] have shown that the coherence the spike train of the FitzHugh-Nagumomodel peaks at an intermediate noise value, in a phenomenon which has been called “coherence resonance”.In other words, the normalized standard deviation R p ≡ q h t p i − h t p i h t p i (8)should have a minimum as a function of the noise intensity. This is precisely what we observe in our circuitwhen V DC (= − .
826 V) is close to the Hopf bifurcation ( V Hopf = − .
82 V), as displayed in Fig. 4d. Notethat R p close to zero means that the time series is approximately periodic.For small noise amplitudes ( V noise ∼
50 mV, or A ∼ O (1) in Fig. 4d), spikes are sparse and R p approaches unity. This suggests a Poisson process in which the interspike interval distribution approachesan exponential P ( t p ) = re − rt p , (9)where r is time rate constant. This Poisson limit is interesting because it is observed in different neuronalpreparations [Dayan & Abott, 2001; Petracchi et al. , 1995], so we performed a detailed statistical analysisof the small V noise regime. -4 -3 -2 -1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 D ( t ) t (s) r=10.0+-0.3 s -1 a) fit e -rt < n > time window: T (s) r=9.060+-0.006 s -1 b)
4. Dynamic range
In this section we study the response of our excitable system to varying input voltage V DC , considering thenoise amplitude V noise constant. Although in real neurons the background noise may have a dependenceon the stimulus, it is a fair approximation to treat the noise amplitude as constant and focus on thedependence on input signal as a control parameter of the dynamics. In what follows, the response of thecircuit is defined as the mean firing rate F measured over a fixed time interval T m . This so-called “ratecoding” is also a longstanding approximation [Adrian, 1926], which seems to fit data in several cases [Koch,1999; Arbib, 2002].For fixed T m and noise amplification A , the response F of our circuit is an increasing function of thestimulus V DC because larger values of V DC amounts to increased excitability, lowering the “effective thresh-old” to noise-induced spike generation (there is no real threshold in type-II excitable neurons [Izhikevich,2007]). Conversely, for fixed V DC , the response F also increases with increasing noise intensity A . Theseresults are shown in Fig. 6a, where we plot (for different noise intensities) the responses F ( V DC ) of ourexcitable circuit with a 1 nF capacitor. This choice sets the time scale of the neuron in the millisecondrange (i.e. that of biological neurons). Note that in the absence of noise ( A = 0) the response is null up to F ( / s ) V DC (V)a) Hopfbifurcation
A = 2000A = 1500A = 1000A = 500A = 0 0 1000 2000 3000 4000 5000-9.4 -9.2 -9 -8.8 -8.6 -8.4 -8.2 -8 -7.8 -7.6 F ( / s ) V DC (V)b) Hopfbifurcation
A = 2000A = 1500A = 1000A = 500A = 0 0 50 100 150 200 0 0.02 0.04 0.06 0.08 F ( / s ) V DC -V (V)c) ∆ = 6.62 dBV V V Hopf
A = 1000A = 0 2 3 4 5 6 7 8 9 10 0 500 1000 1500 2000 2500 ∆ ( d B ) Ad) C = 1 nFC = 50 pF
Fig. 6. Experimental response curves F ( V DC ) measured at different values of the noise amplification A . Supply voltages V a = 12 V = − V b . a) C = 1 nF ( φ = 5 × − and T m = 10 s). b) C = 50 pF ( φ = 0 .
01 and T m = 0 . C = 1 nF (A=1000), and relevant parameter for calculating the dynamic range. d) Dynamic range as function of noiseamplification for C = 50 pF (black squares) and C = 1 nF (white circles). ctober 24, 2018 3:50 medeiros-ijbc-2011 REFERENCES the Hopf bifurcation (so the lowest curve in Fig. 6a is similar to Fig. 2c).Results in Fig. 6b correspond to a circuit with a 50 pF capacitor. This single change renders a muchfaster circuit, now operating in the microsecond range, but with its dynamical features otherwise preserved.This has potential applications, because a faster circuit requires shorter measurement intervals T m (=0.2 sin our example) for a reliable estimation of the firing rate.Given a response curve, we can calculate its dynamic range, which roughly speaking corresponds tothe range of stimulus intensity that the firing rate can “appropriately code”. Measured in decibels, this isarbitrarily defined as [Rospars et al. , 2000; Copelli & Kinouchi, 2005]∆ ≡
10 log (cid:18) V ∗ . V ∗ . (cid:19) , (11)where V ∗ x ≡ V x − V are measured relative to the voltage V at which the response becomes non-zero and F ( V x ) = xF max (0 ≤ x ≤ , (12)where F max is the firing rate at the Hopf bifurcation. In words (see Fig. 6c), ∆ measures the range ofstimulus V DC which are neither too small ( V DC < V . ) to go undetected nor too close ( V DC > V . ) to theautonomous oscillations that emerge at V Hopf .As shown in Fig. 6d, the dynamic range is a rather robust feature of our excitable circuit: itchanges little as the noise intensity is varied, regardless of the time scale at which it operates. Inboth cases, ∆ ≃ ≃
10 dB for olfac-tory sensory neurons [Rospars et al. , 2000], ∆ ≃
14 dB for retinal ganglion cells [Deans et al. , 2002;Furtado & Copelli, 2006]) than results obtained theoretically for discrete models of excitable elements(∆ ≃
14 dB in [Furtado & Copelli, 2006] and ∆ ≃
19 dB in [Assis & Copelli, 2008]).
5. Concluding remarks
In summary, we have presented an excitable electronic circuit whose simplicity allows for scalability andaccurate mathematical modeling. Its dynamical equations lead to time series which quantitatively reproduceexperimental results without fitting parameters.In addition, we have shown that the introduction of noise from a simple analog noise generator atthe input of the circuit produces variable spike trains. The statistics of the interspike intervals is shownto exhibit coherence resonance. Furthermore, by analyzing long time series under low noise intensity, thespike trains were shown to behave as a Poisson process, like some biological neurons.In the excitable regime, with fixed noise amplitude, the firing rate response of the system to a V DC input – the stimulus – was shown to have a dynamic range of about 6 dB, which is also comparable tosome biological sensory neurons. Together with its scalability, these properties render the system a potentialbuilding block for artificial sensors based on collective properties of excitable media.
6. Acknowledgments
BNSM, MC and JRRL acknowledge financial support from Brazilian agencies CNPq, FACEPE, CAPESand special programs PRONEX, PRONEM and INCEMAQ. GBM acknowledges support from NIH. It isa pleasure to thank Hugo L. D. S. Cavalcante for enlightening discussions during the preparation of thiswork, as well as Marcos Nascimento for technical support.
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