Signal anticipation and delay in excitable media: group delay of the FitzHugh-Nagumo model
aa r X i v : . [ q - b i o . N C ] J a n Signal anticipation and delay in excitable media: group delay of theFitzHugh-Nagumo model
Akke Mats HoubenJanuary 5, 2021
Abstract
An expression for the group delay of the FitzHugh-Nagumo model in response to low amplitude input isobtained by linearisation of the cubic term of the volt-age equation around its stable fixed-point. It is foundthat a negative group delay exists for low frequencies,indicating that the evolution of slowly fluctuating sig-nals are anticipated by the voltage dynamics.The effects of the group delay for different types ofsignals are shown numerically for the non-linearisedFitzHugh-Nagumo model, and some observations onthe signal aspects that are anticipated are stated.
A neuronal spike is a fast transient event, lastingtypically around 1 ms, but transmission of spikesbetween pairs of neurons can take between 1 and100 ms [1, 9, 40, 39], and the typical timescale of sub-threshold changes in the membrane potential of sev-eral neuron models, as fitted to experimental data,is usually set from 10 to 100 ms [10]. Thus we canconclude that there must exists a non-negligible de-lay between a neuron receiving a stimulation and theresultant action potential arriving at another neuron.This delay seems at odds with the well-known ob-servations that behavioural or electrophysiological re-sponses (which both seem to involve large numbersof neurons) generally occur in the order of just a fewhundred milliseconds [e.g. 42, 43]. It is thus not sur-prising that rapid signal propagation between neu-rons and within neuronal networks has received a lot of attention [e.g. 11, 14, 29, 30, 31, 33, 41]. It hasbeen proposed that dynamical systems with certainproperties can anticipate their input signals. Some(excitable) dynamical systems, such as neurons [e.g.8, 32] and neural systems [25], have been shown toanticipate aspects of their inputs through so-calledanticipated synchronisation [46], which for excitablesystems appears through canard solutions to the sys-tem [20]. Alternatively, model neurons with spe-cific properties [47], as well as some electrical sys-tems [e.g. 24, 26, 28] and certain active media [e.g.4, 5, 7, 15, 34, 36, 37, 48, 50], are able to anticipatethe transmission of aspects of specific signals throughthe existence of a negative group delay or unexpect-edly high group velocities.In this article, I show that the dynamics of awidely used model for excitable media, the FitzHugh-Nagumo model [13, 19, 27], near its stable fixed point¯ p (¯ v, ¯ w ), possesses a frequency band with a negativegroup delay. Meaning that the dynamics of the modelanticipate ‘sub-threshold’ inputs of signals with cer-tain characteristics within this frequency band. Achain of these models then seemingly respond to thesetypes of signals before the first in the chain is goodand well stimulated.In what follows, first an illustration of negativegroup delay is given (section 2), and an expressionfor the group delay of the FitzHugh-Nagumo modelfor low-amplitude input is obtained (section 3). Fol-lowing (section 4) the effects of the group delay aredemonstrated for several types of signals both numer-ically and conceptually.1 Negative group delay
Group delay τ ( ω ) is defined as the additive inverseof the derivative of the phase-response e H ( ω ) of asystem: τ ( ω ) := − ddω e H ( ω ) , and determines the time-delay of the amplitude en-velope of inputs at each frequency ω [2, 26, 50].To understand the effect of a group delay, considera signal f ( t ) = g ( t ) e i ω c t , consisting of a sinusoid withfrequency ω c , modulated by a, relative to the input,low-frequency envelope g ( t ), being passed through afilter e H ( ω ) := | e H ( ω ) | e i φ ( ω ) . For simplicity assumethe filter has a flat unit amplitude response | e H ( ω ) | :=1. The spectrum of f ( t ) passed through the filter is e f ( ω ) e H ( ω ) = e g ( ω − ω c ) e iφ ( ω ) . Since the modulation of the envelope is much slowerthan that of the sinusoidal signal, the spectrum of G ( ω ) is, relative to the signal, narrow around ω = 0.So, the output signal will contain a narrow band offrequencies, around ω = ω c . Approximating φ ( ω ) byits Taylor series at ω c up to first order, and substitut-ing Ω = ω − ω c gives for the inverse Fourier transform( f ∗ H )( t ) = e i ( ω c t + φ ( ω c )) Z ∞−∞ e g (Ω) e i φ ′ ( ω c )Ω e i Ω t d Ω , leading to, by the definition of group delay,( f ∗ H )( t ) = g ( t − τ ( ω c )) e i ( ω c t + φ ( ω c )) . Thus the envelope g ( t ) of a signal at frequency ω c isshifted by an amount of τ ( ω c ). Systems where τ ( ω )is negative for a range of ω are said to contain a nega-tive group delay, and the output of these systems willanticipate the envelope of signals within this band[2, 26, 50]. For a broad-band signal f ( t ) filtered by a narrow-band filter h ω c ( t ) with a spectrum centered at ω c , we can show that the group-delay instead shifts the com-plete signal ( h ω c ∗ f )( t ). Passing this signal throughthe same filter H ( t ) as before gives(( h ω c ∗ f ) ∗ H )( t ) = Z ( e h ω c · e f )( ω ) e i φ ( ω ) e i ωt dω. Assuming that e h ω c ( ω ) decays rapidly at both sidesof ω c , we can use the same linear approximation asbefore, leading to e i [ ω c t + φ ( ω c )] Z ( e h ω c · e f )(Ω + ω c ) e i φ ′ ( ω c )Ω e i Ω t d Ω , which shows that this results in a time-shift and scal-ing of the filtered signal:(( h ω c ∗ f ) ∗ H )( t ) = e i φ ( ω c ) ( h ω c ∗ f )( t − τ ( ω c )) . So for signals filtered narrowly around a frequency ω c it is the complete signal that is shifted by an amount τ ( ω c ). The FitzHugh-Nagumo model [13, 27] describes thedynamics of the membrane potential v of a neuronalongside a recovery variable w : dvdt = v − v − w + Idwdt = a ( v + b − cw ) , (1)with three parameters: a , determining the timescaleof the dynamics of w ; an offset b ; and c which influ-ences the slope of the w -nullcline and decay rate of w .The fixed point ¯ p (¯ v, ¯ w ) of (1) can be easily found bysolving the cubic equation ¯ v +(1 /c − v + b/c = I andplugging the resulting value for ¯ v into the equation¯ w = (¯ v + b ) /c . For low-amplitude input I , if the fixed-point is stable we can approximate the cubic term v by its Taylor series at ¯ v , allowing the equation for thevoltage evolution of (1) to be approximated by that2 | H ( ω ) | ω (10 rad·s -1 )b=0.7b=0.6b=0.9 (a) | e H ( ω ) | -1.5-1-0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 a r g H ( ω ) (r ad ) ω (10 rad·s -1 )b=0.7b=0.6b=0.9 (b) e H ( ω ) Figure 1:
Transfer function e H ( ω ) of the linearisedvoltage variable of the FitzHugh-Nagumo model. a)theoretical magnitude response (black lines) and theoutput spectrum obtained by numerical integrationof the non-linear equations (grey connected symbols).b) theoretical phase response of linearised equations.Both panels show curves for different values for b of an approximated membrane potential x = ¯ v + ǫ ,with governing equation dxdt = x − ¯ v − ¯ v ( x − ¯ v ) − w + I + O ( ǫ ) (2)which is a linear first order differential equation. Itis then straightforward to obtain the spectrum of thedynamics of x : e x ( ω ) = e I ( ω ) + DC( ω )¯ v − i ω + aac + i ω , (3)with direct term DC = [ ab/ ( ac + i ω ) + (1 − / v ] δ ( ω ). If we ignore the DC term, dividing e x by e I leads to the transfer function e H ( ω ) of v close to¯ v . Figures 1a and 1b show, respectively, themagnitude- and phase-transfer functions of the lin-earised voltage variable (solid lines) and the spectrumof the original non-linear equation obtained numeri-cally (dashed lines) for different parameters.The phase-spectrum is the argument of thetransfer-function e H ( ω ) = tan − (cid:18) aω/ ( a c + ω ) − ω ¯ v − a c/ ( a c + ω ) (cid:19) , whose additive inverse differentiated with respect to ω gives the group delay τ ( ω ) = − ddω tan − (cid:18) aω/ ( a c + ω ) − ω ¯ v − a c/ ( a c + ω ) (cid:19) = − a c − a ((¯ v − ac ) ω ( a c + ω ) + a (¯ v − − a ca c + ω − (¯ v − ωa/ ( a c + ω ) − ω ] + [(¯ v −
1) + a c/ ( a c + ω )] . (4)In order for (4) to be negative we need Aω + Bω + C < , (5)where A := (¯ v − B := (¯ v − a c + a ) + 3 a cC := (¯ v − a c − a c ) + a c − a c. The l.h.s. of (5) has a real positive root only if B − AC ≥ ω = vuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ B − AC − B A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Using the slope of (5) at this zero-crossing ddω Aω + Bω + C (cid:12)(cid:12)(cid:12)(cid:12) ω = ω = 4 Aω + 2 Bω , noting that 0 < ω < a and that for stable fixed-points ¯ v − ac >
0, we obtain the result that (2)has a frequency band with negative group delay for ω < ω .The group delay is maximally negative for ω =0 and increases monotonically to ω = ω , meaningthat the envelope of signals with frequencies 0 < ω <ω are transmitted with a negative delay. For highfrequencies (4) tends to the limitlim ω →∞ τ ( ω ) = 0 + , thus high frequencies are transmitted without signif-icant delay or anticipation. In fact, for the canonicalcase in which the timescales of v and w are suffi-ciently separated, a ≪
1, the group delay approachesthis limit fast for ω >
1. The group delay is thenmaximal for ω < ω <
1. Figure 2 shows the groupdelay function τ ( ω ) for some different parameters.3 τ ( ω ) ( m s ) ω (10 rad·s -1 )b=0.7b=0.6b=0.9 Figure 2:
Group delay function τ ( ω ) of the voltagevariable of the linearised FitzHugh-Nagumo modelfor different values for parameter b Group delay affects the envelope of band-limited sig-nals, so the types of signals for which the group delayof the membrane potential will have an effect are themodulations of ‘constant’ band-limited signals. Thelinear approximation of φ ( ω ) at ω = ω c during theillustration of the group delay implies, as stated be-fore, that the frequency spectrum of the modulationneeds to have a low-pass characteristic, so that thespectrum of the modulation is narrow around ω = 0.In the following the anticipation of different typesof signals by the voltage variable of the non-linearisedmodel neuron are shown numerically. In order todifferentiate between the effects of the phase delay − φ ( ω ) /ω and the group delay the simulations arecarried out with a chain of 17 model neurons, thusleading to a total expected group delay of 17 τ ( ω ). Ineach run the first in the chain of neurons receives aninput I ( t ), and the subsequent neurons receive aninput directly from the membrane potential of each previous neuron I i ( t ) = η [ v i − ( t ) − v ] , for i = 1 , , , . . . with coupling strength η . For each simulation allneurons have parameter values a = 0 . b = 0 . c = 0 .
8, leading to a negative group delay for fre-quencies below ω / (2 π ) = ν ≈ .
14 Hz (c.f. fig.2),and η = 0 . | H ( ω ) | − .Since τ ( ω ) is not flat for ω <
1, considerable fre-quency smearing is expected even for narrow-bandsignals in this range. Therefore, in the following nu-merical examples it is not expected that the mag-nitude of the observed time-shift corresponds abso-lutely to the theoretical prediction but, as will beshown, the time-shifts correspond qualitatively to theshape of τ ( ω ) and in most cases agree well on themagnitude of the time-shift as well. The classical way to show the effects of group delay isto use wave pulses: sinusoidal signals of different fre-quencies modulated by a windowing function. Thesesignals would correspond to short oscillatory bursts,which could indicate or establish transient coordina-tion of the activity of otherwise independent elements[21, 38, 49], which is proposed to underly processesin the brain [e.g. 6, 16, 45]. In a more general sensesignals of this type are amplitude modulated signals,in which the amplitude of a (high frequency) carriersignal is modulated by a lower frequency signal whichis to be transmitted.The windowing function used in the following is aGaussian pulse g ( t ) = e − αt of width σ , which has a spectrum of the form e g ( ω ) = r πα e − ω α , which is centered at ω = 0 and whose magnitude | e g ( ω ) | has a fast roll off depending on α . Thus forwide enough Gaussian pulses, leading to small valuesfor α , the linear approximation of the phase-spectrumshould hold.4 m p li t ude I (t) a m p li t ude v M-1 (t)-1000 -500 0 500 1000 en v e l ope po w e r time (ms) I (t)v M-1 (t) (a) ω c = ω / a m p li t ude I (t) a m p li t ude v M-1 (t)-1000 -500 0 500 1000 en v e l ope po w e r time (ms) I (t)v M-1 (t) (b) ω c = 2 ω Figure 3:
Anticipation and delay of Gaussianwave pulses with carrier frequency within the neg-ative and positive, respectively, group delay band bya chain of model neuronsThe top panels of fig.3 show the input wave pulseswith carrier frequencies of ν c = ν / ≈ .
57 Hz (left-panel) and ν c = 2 ν ≈ .
28 Hz (right-panel), falling,respectively, into the negative group delay band and aband near the maximal group delay of the membranepotential (c.f. Fig.2). The middle panels show themembrane potential of the last neuron in a chain of17 cascaded model neurons, in which the first neuronreceived the wave pulse as in the first panel I ( t ) = e − α ( t − t ) A sin ( ω c t )at time t = t with carrier frequency ω c and ampli-tude A =1 × − The bottom panels of Fig.3 show the amplitude en-velope of the input signal (solid lines) and that of thelast neuron (dashed lines), measured as the output ofa strong lowpass filter, for the two different signals.In the membrane potential of the last neuron, aclear shift forward in time of the signal envelope isvisible for signals in the negative group delay band(bottom left panel), in agreement with the group de-lay predicted by (4), whereas the envelope of the sig-nal in the positive group delay band is delayed (bot-tom right panel).The time-shift versus pulse-carrier frequency isshown by fig.4 which shows the time lag (y-axis) ofthe peak in the correlation of the membrane poten-tial of the first neuron, receiving an input Gaussian -10-8-6-4-2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 de l a y ( m s ) ω c (10 rad·s -1 )simulation τ ( ω ) Figure 4:
Peak correlation time versus ω c forGaussian wave pulses. Numerical (connected dots)alongside theoretical (solid black line) group delaypulse with carrier frequency ω c (x-axis), with themembrane potential of the last of the chain of 17neurons. It is visible that the numerical results (con-nected dots) agree well with the expected group delay τ ( ω c ) (solid black line). Pure sinusoidal signals, even though they allow sim-ple analyses, are rare. More commonly, biologicaland physical signals are considered to be noisy [e.g.3, 12, 22, 35, 44]. In the following it is shown,as anticipated, that the membrane potential of themodel neuron also anticipates the fluctuations ofband-limited white noise input.An ideal zero-mean Gaussian white noise ξ ( t ) withvariance σ has a flat, wide-band, power spectral den-sity | e ξ ( ω ) | = R σ δ ( t ) e − i ωt dt = σ . In the following ξ ( t ) is a normal Gaussian noise, thus σ = 1. Thissignal will be filtered by a Gaussian filter centered ata frequency ω c and a narrow band-width determinedby 0 < α ≪
1. Thus the first neuron receives an5 m p li t ude I (t) a m p li t ude v M-1 (t)-1000 -500 0 500 1000 en v e l ope po w e r time (ms) I (t)v M-1 (t) (a) ω c = ω / a m p li t ude I (t) a m p li t ude v M-1 (t)-1000 -500 0 500 1000 en v e l ope po w e r time (ms) I (t)v M-1 (t)21ms (b) ω c = 2 ω Figure 5:
Anticipation and delay of band-limited noise pulses with pass-band within a neg-ative and positive group delay band, respectivelyinput with spectrum e I ( ω ) = A e − ( ω − ωc )24 α e ξ ( ω ) . Such narrow band filtering of sub-threshold inputscan arise due to signal transmission with heteroge-neous transmission delays between neurons [18].The left column of fig.5 shows the results of thechain of model neurons being driven by a band-limited noise with center-frequency ν c = ν / ≈ .
57 Hz, and α =1 × − falling within the negativegroup delay band. The top panels show a section ofone input signal (top most panel) together with themembrane potential of the last neuron in the chain(second to top panel) in response to that signal. Thebottom most panel shows the envelope of the input(solid line) and that of the membrane potential of thelast neuron (dashed line). The right panel shows thesame, but for an input with ν c = 2 ν ≈ .
28 Hz.Fig.6 shows the time lag of the first peak in the cor-relation function between a filtered noise input withcenter-frequency ω c (x-axis) and the membrane po-tential of the model neuron receiving the input (con-nected symbols), for different filtering bandwidths α .The solid black line shows the theoretical group delay τ ( ω c ). -10-8-6-4-2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 de l a y ( m s ) ω c (10 rad·s -1 ) α = 5e-4 α = 1e-4 α = 1e-5 τ ( ω ) Figure 6:
Peak correlation time versus ω c forfiltered noise input. Numerical results (connectedsymbols) for different filtering band-widths, alongsidetheoretical group delay (solid black line) In the foregoing section the noise considered was sta-tionary, however it is likely that in biological andphysical settings the noise sources evolve over time.In addition, modulations of stationary noise sourcescan serve as a form of signalling through a medium.Such as mean- and variance-modulation as proposed‘communication channels’ for neurons and neural net-works [e.g. 17, 23].Considering a noise signal subjected to slow mean-and variance-modulation, g µ ( t ) and g σ ( t ), f ( t ) = g µ ( t ) + g σ ( t ) ξ ( t ) , with either the result f ( t ) or the input noise ξ ( t )being passed through a band-pass filter as before.Clearly in each case the noise signal will be affectedby the group delay, if the pass-band falls within a fre-quency range exhibiting a group delay. It is also ap-parent that in both cases the mean-modulation g µ ( t )will not be affected by the group delay, and thus willbe passed without delay or advance.6he variance-modulation however will be affected,but only in case of the noise signal being band-limitedbefore the variance modulation, which is easily under-stood from the fact that the power of g σ ( t ) is focussedoutside of the pass band of the band-limited filter andthus will only serve as an amplitude modulation tothe higher frequency noise signal. In this paper it is shown that the dynamics ofthe voltage variable of the FitzHugh-Nagumo modelposesses a negative group delay for slowly fluctuatinginputs, by finding an expression for the group delay ofa linear approximation of the governing equation forthe voltage variable at the fixed point of the modelsystem.The effects of the group delay are demonstratednumerically for different types of signals and signalmodulations and it is shown that a chain of uni-directionally coupled model neurons anticipates cer-tain aspects of inputs if the input consists of a carriersignal with a frequency falling within the negativegroup delay band.
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