neural adaptation facilitates oscillatory responses to ...neural adaptation facilitates oscillatory...

J Comput Neurosci (2011) 31:73–86DOI 10.1007/s10827-010-0298-4

Neural adaptation facilitates oscillatory responses to staticinputs in a recurrent network of ON and OFF cells

Jeremie Lefebvre · Andre Longtin ·Victor G. LeBlanc

Received: 15 July 2010 / Revised: 6 October 2010 / Accepted: 26 November 2010 / Published online: 18 December 2010© Springer Science+Business Media, LLC 2010

Abstract We investigate the role of adaptation in aneural field model, composed of ON and OFF cells,with delayed all-to-all recurrent connections. As ex-ternal spatially profiled inputs drive the network, ONcells receive inputs directly, while OFF cells receive aninverted image of the original signals. Via global anddelayed inhibitory connections, these signals can causethe system to enter states of sustained oscillatory activ-ity. We perform a bifurcation analysis of our model toelucidate how neural adaptation influences the abilityof the network to exhibit oscillatory activity. We showthat slow adaptation encourages input-induced rhyth-mic states by decreasing the Andronov–Hopf bifurca-tion threshold. We further determine how the feedbackand adaptation together shape the resonant propertiesof the ON and OFF cell network and how this affectsthe response to time-periodic input. By introducing anadditional frequency in the system, adaptation altersthe resonance frequency by shifting the peaks wherethe response is maximal. We support these results withnumerical experiments of the neural field model. Al-though developed in the context of the circuitry of theelectric sense, these results are applicable to any net-work of spontaneously firing cells with global inhibitoryfeedback to themselves, in which a fraction of thesecells receive external input directly, while the remain-ing ones receive an inverted version of this input viafeedforward di-synaptic inhibition. Thus the results are

Action Editor: Brent Doiron

J. Lefebvre (B) · A. Longtin · V. G. LeBlanc150 Louis Pasteur, Ottawa, ON, Canada K1N 6N5e-mail: [email protected]

relevant beyond the many sensory systems where ONand OFF cells are usually identified, and provide thebackbone for understanding dynamical network effectsof lateral connections and various forms of ON/OFFresponses.

Keywords Neural field · Delayed feedback ·Sensory inputs · Oscillations · ON and OFF cells ·Bifurcations · Frequency tuning

1 Introduction

The behavior of neural systems is governed by a com-bination of circuitry and cellular attributes. Amongstthese, spike frequency adaptation is found in almost allneurons, where it is thought to influence the processingof neural information mediated by action potentials.Adaptation corresponds to a stereotyped decrease infiring rate after prolonged stimulation, as the cell ha-bituates to steady input currents. It is thought to playa particularly important role in sensory systems. Thereit can alter neuronal firing patterns in order to directthe system‘s response towards given stimulus attri-butes (Benda et al. 2001; Kim and Rieke 2001; Wanget al. 2003; Gollisch and Herz 2004; Benda et al. 2005;Gabbiani and Krapp 2006) or to tune neural sensitivityto stimulus intensity (Sobel and Tank 1994). Adap-tation has further been shown to control repetitivefiring (Prescott et al. 2006) and influences both timeand rate coding properties (Prescott and Sejnowski2008). Various mechanisms underlying adaptation havebeen identified. Steady neuron firing can activate slowpotassium currents (Storm 1990), which may also becalcium-dependent (Sah and Davies 2000), resulting in

74 J Comput Neurosci (2011) 31:73–86

firing rate decay following a step input. Adaptationhas also been linked in other cases to the inactivationof slow sodium currents (Kim and Rieke 2003). The-oretical studies on Integrate-and-Fire models as wellas conductance-based models have reproduced exper-imental recordings of adapting behavior (see Bendaet al. (2001), Liu and Wang (2001), Benda and Herz(2003) and references therein).

The goal of this paper is to investigate how adapta-tion shapes the frequency tuning of cells and stimulus-induced network oscillations in a realistic context ofsensory feedback circuitry involving adaptive ON andOFF cells. We draw our main motivation for combiningfeedback, ON/OFF populations and adaptation fromstudies of the weakly electric fish (Apteronotus lep-torhynchus). Adaptation has been studied there bothat the level of the primary receptor known as the P-unit electroreceptor (which we do not focus on here),as well as at the level of the post-synaptic popula-tion of pyramidal cells. P-unit adaptation is very rapid(tens of milliseconds) and has been shown to influencethe frequency-dependent encoding of electrosensoryinputs (Xu et al. 1996; Doiron et al. 2004). This adap-tation further enables the separation of fast transientstimuli, related to communication signals, from sloweroscillatory signals arising from the proximity of two fish(Benda et al. 2005). This adaptation further participatesin the appearance of input-induced states of synchrony(Benda et al. 2006), allowing transitions among P-unitsfrom states of synchrony to desynchrony and vice-versa due to rapid communication signals (Whittingtonet al. 1995). Each P-unit axon then trifurcates, witheach of the three processes reaching pyramidal cells inone of three topographic maps of the electrosensorylateral line lobe (ELL). Cells especially in superficiallayers of the ELL also exhibit adaptation which shapestheir temporal filtering properties for oscillatory inputsthat arise naturally during an encounter of two fish(Mathieson and Maler 1988; Mehaffey et al. 2008;Krahe et al. 2008). In fact adaptation becomes fasteras one moves from central to lateral maps, which moti-vates the study here across adaptation time scales. Themechanism of this adaptation is not known but doesseem to depend on calcium (Dr. Len Maler, personalcommunication).

It is the adaptation exhibited by these latter pyrami-dal cells that is of interest in our paper because theyare involved in recurrent circuitry with other nuclei—asopposed to receptors which are involved only in feed-forward circuitry. In the visual system, the thalamo-cortical loop has similar properties and exhibitsstructures that also possess ON and OFF cells. We notethat adaptation is in fact a form of negative feedback,

and as such can interact with—and even mimic—otherforms of feedback caused by network circuitry. Recentdynamical studies on large scale nets have shed lighton the role of adaptation in the generation and stabilityof spatially localized patterns like breathers (localizedtime-periodic bumps of activity) and traveling waves(Curtu and Ermentrout 2004; Folias and Bressloff 2005;Kilpatrick and Bressloff 2010). Other studies demon-strated its impact on network oscillations (Crook et al.1998; Ermentrout et al. 2001; van Vreeswijk and Hansel2001) in the form of enhanced synchronization.

Of further interest is the fact that oscillatory statescan appear in sensory pathways as a consequence ofsensory inputs with sufficiently high spatial coherenceand/or spatial binding (Gray and Singer 1989; Borgersand Kopell 2003; Borgers et al. 2008). In the weaklyelectric fish such oscillations are associated with tempo-rally random stimuli of large spatial correlation (such asother animals) and relies on delayed feedback (Doironet al. 2003, 2004; Marinazzo et al. 2007; Lindner et al.2005). Delayed feedback inhibition common to all cellsoften underlies oscillatory activity in the brain as itcompetes with excitatory feedback (see e.g. Pauluiset al. (1999), Pauluis (2000), Borgers and Kopell (2003),Dhamala et al. (2004), Borgers et al. (2008), Brandt andWessel (2007) and references therein). Also, frequencytuning effects have been observed in the electric fishthat change with the spatial configuration of the stim-ulus, i.e. on its local versus global geometry (Bastianet al. 2002; Doiron et al. 2003; Chacron et al. 2005).These are due in part to cellular and circuit properties(Chacron et al. 2005; Krahe et al. 2008). It is known forexample that a step increase in stimulus contrast causesan increase followed by a decrease in ELL firing, i.e. byadaptive behavior rather than oscillatory behavior.

These studies naturally lead to the question of howadaptation interacts with spatio-temporal stimuli thatlead to oscillatory dynamics. Does the presence ofadaptation increase or decrease the propensity for aninhibitory recurrent network to oscillate in responseto spatially correlated inputs? How does adaptationinfluence frequency tuning in the presence of recur-rent inhibition? Cells in the ELL further display eithersimple ON or OFF behavior: ON (OFF) cells encodepositive (negative)-going fluctuations of input signals.These signals occur as modulations of the amplitude(and sometimes frequency for communication calls) ofthe carrier oscillation emitted by the fish known asthe electric organ discharge (EOD). ON cells (knownas E cells) thus increase their firing rate when theamplitude of the EOD increases, and vice-versa for theOFF cells (known as I cells) (Berman and Maler 1998,1999; Maler et al. 1991). The incorporation of multiple

J Comput Neurosci (2011) 31:73–86 75

neural populations is important to properly account fornetwork activity and receptive field geometry and isstill at the forefront of work in theoretical neuroscience(Wilson and Cowan 1972; Golomb and Ermentrout2001; Laing and Coombes 2006; Blomquist et al. 2005).In earlier work inspired by the electrosensory circuitry,we have shown how ON and OFF populations inter-act with delayed and non-delayed recurrent connec-tions to generate oscillations triggered by static stimuli(Lefebvre et al. 2009, 2010). These studies, which usedneural field formulations as well as stochastic Integrate-and-Fire neurons, did not consider the issue of cellularadaptation; in fact the dynamics of networks with bothmultiple populations and adaptation is a general openquestion.

We note that the interplay of ON and OFF path-ways also plays a fundamental role in vision fromretina onwards (Kandel and Schwarz 1983; Gollisch andHerz 2004) as well as in audition (Robin and Royer1987; Scholl et al. 2010) and other senses. The simpleON/OFF dichotomy described above, where each pop-ulation responds preferentially to one polarity of thestimulus, is commonly present (Gabbiani 1996). Yet thecircuitry and physiology, often involving many typesof ON/OFF cells and complex network interactions, isfar from clear and is slowly being elucidated (Gollischand Herz 2004; Gollisch and Meister 2008; Liang andFreed 2010; Scholl et al. 2010). Here we focus on thissimple type of ON/OFF behavior as opposed to otherforms, such as that where both the onset and offset ofa stimulus both cause firing rate increases (Scholl et al.2010) (in the simple dichotomy illustrated above, ONand OFF populations would have inverted responseswith respect to one another for both onset and offset).

In contrast, ON/OFF circuitry (known as E/I cir-cuitry) has been worked out for weakly electric fish,and can be summarized as follows: ON and OFF cellsshare common P-unit afferents, but the OFF pathwayincludes an interposed inhibitory interneuron, causingthe OFF response to be inverted (Berman and Maler1998, 1999). Electroreception thus offers a relativelysimpler sensory system, both anatomically and phys-iologically, in which to investigate the role of adap-tive ON/OFF cells involved in recurrent circuitry. Sucha study can then provide the dynamical backbonefor more complex systems and forms of ON/OFF re-sponses. It is important to note that earlier modelingstudies of oscillations in ELL assumed only one popu-lation was at work (the ON population (Doiron et al.2003, 2004; Marinazzo et al. 2007; Lindner et al. 2005)).

In this paper, we address the following questions:how does adaptation influence the oscillatory responsethreshold in networks of ON and OFF cells? Are global

oscillations as common when adaptation is included?How does the combination of adaptation and feed-back shape frequency tuning of cells embedded in thenetwork? Our work here builds on our previous re-sults about ON and OFF cells and delayed feedbackwithout adaptation. We investigate how adaptationmay underlie stimulus-induced oscillations in recur-rent networks. To demonstrate this, we compare theAndronov–Hopf bifurcation scenario for the cases withand without adaptation. We thus expand the stabilityanalysis around steady activity states in a recurrentmodel that now includes adaptation dynamics. We willhighlight the effect of adaptation on the stability ofinput-induced limit cycles. We will also study howsuch a model responds to time-periodic input, and howadaptation shapes the resonance curve by introducingadditional time-scales in the system.

In Section 2, we describe the architecture of ourON/OFF network and we show how stable oscillationsappear as a result of spatially localized stimulation. InSection 3, we perform a stability analysis of our model,incorporating adaptation, where we compare the insta-bility point between the cases where adaptation is andis not present. There, our study has been made moreanalytically tractable by assuming identical adaptationdynamics for the ON and OFF cells. We further lookat the impact of adaptation on the system’s steadystates, to determine how external input then interactwith the feedback. Lastly, in Section 4, we investigateand compare the effects of adaptation and feedbackon the amplitude of the cells response to time-varyinginputs. We further motivate those results by comparingthe resonance curves with those obtained with a noisyIntegrate-and-Fire net.

2 Model

Our model is based on electroreception, but is generalenough to apply to other senses (Fig. 1). We describethe evolution of the neural activity u(x, t), correspond-ing to the mean somatic membrane potential of a sub-network of the whole network located at position xalong a one-dimensional spatial domain �. This activityis further segregated into that of ON and OFF popula-tions uon,of f . External sensory signals I(x, t) propagatein a parallel fashion from the first receptors (not mod-eled explicitly) up to the “sensory” layer of pyramidalcells in the ELL, and excite and/or inhibit the localpopulations by altering their activity level. Further,recurrent connections allow the activity of the cells topropagate to higher brain centers (mainly area NP inthe weakly electric fish (Berman and Maler 1999)).


Fig. 1 Network architecture of our model. It is inspired from theELL of the weakly electric fish. The sensory layer, built of anequal number of ON and OFF cells, receives external sensoryinputs with direct (ON) and inverted (OFF) polarity. The popula-tions then project to higher centers, which accumulate the activitydistributed across the network (sigma symbol). The recurrentconnections allows the accumulated activity component to besent back to all the cells in the sensory layer with some time lag τ

For simplicity the activity there is summed across thenetwork and then fed back to all the initial sites globallyand with inhibitory polarity. This component of the cir-cuit involves a significant processing and propagationtime lag modeled with a fixed delay τ . The fields uon

and uof f obey the dynamics

(1 + a−1∂t

)uon(x, t) = −A(t − τ) + I(x, t)

(1 + a−1∂t

)uof f (x, t) = −A(t − τ) − I(x, t), (1)

where a is the rate constant of the exponentialsynapses with response function η(t) = ae−at, and Ais the delayed inhibitory feedback connection, corre-sponding to the accumulation of activity of each unitacross the network:

A(t − τ)

=∫

�

dy[αon f (uon(y, t − τ)) + αof f f

(uof f (y, t − τ)

) ].

Here, αon and αof f are the relative proportions ofON and OFF cells in the network, which have beenboth set to 0.5. The function f is a sigmoidal firing ratefunction defined by f (u) = (1 + exp(−β(u − h)))−1 fora gain of β and an activation threshold h. The feedbackgain β is fixed to 25 troughout the analysis.

The ELL in the weakly electric fish exhibits a similararchitecture to the one used in this model. Indeed, veryfew lateral connections exist within it, such that most

of the processing is performed by the means of feed-back connections from higher muclei. Previous studies(Lefebvre et al. 2009, 2010) demonstrated that thismodel exhibits changes from stable activity equilibriato global oscillations as a result of increasing stimulusamplitude and/or spatial extent, due to the presenceof an Andronov–Hopf bifurcation. Those neural fieldpredictions were further supported by simulations ofa network of noisy Integrate-and-Fire neurons withspatio-temporal forcing, as in Lindner et al. (2005) andDoiron et al. (2003, 2004) for the case where only ONcells were considered. Weaker effects of local non-delayed circuitry within the ELL itself have also beenstudied (Lefebvre et al. 2010), and were shown to be ofno qualitative consequence on the dynamics; such localeffects are thus not modeled here.

Aside from their inhibitory response to positive in-puts, OFF cells in-vivo can fire at a baseline meanrate even in absence of external stimulation, as mayON cells (Robin and Royer 1987; Laing and Coombes2006). In the electric sense, both ON and OFF cellsare spontaneously active because they receive tonicexcitation from various sources. Thus, external inputmodulates the firing activity of both ON and OFF cellsaround this baseline activity. Experimental recordingsin the electrosensory system indicate that the spon-taneous firing rate may even be slightly higher forOFF cells than for ON cells. Our recent results, in thecontext where no adaptation is present, demonstratethat such a significant difference in spontaneous activ-ity between these neural populations can qualitativelyalter the input response of ON/OFF nets (Laing andCoombes 2006). While the inclusion of this activitydifference in our network with adaptation is easy to do(e.g. by adding a bias current to one neuron popula-tion) and would further enhance the connection of ourresults with the physiology of real ON/OFF systems,the analysis would become much more complicated asa function of this asymmetry. Thus, for simplicity, weassume throughout that both neural populations sharethe same baseline activity in the absence of input. Thisallows us to focus more clearly on the influence ofadaptation on the resonance and oscillatory propertiesof such nets. Also, our results are generally applicableto any network of spontaneously firing cells with globalinhibitory feedback to themselves, in which a fractionof these cells receive external input directly, while theremaining ones receive an inverted version of this inputvia feedforward di-synaptic inhibition.

Without adaptation, observed global oscillations canemerge as network responses to spatially distributedsignals. Figure 2 shows the response of ON and OFFpopulations to an input of the form I(x, t) = Io �= 0


Fig. 2 Oscillatory responseof ON (left panel) and OFF(right panel) populations to aspatially localized pulse.Parameters are a = 1,τ = 1.4, h = 0.1 and� = [0, 1]. The input has anamplitude Io = 0.3 andpossess a spatial width� ≡ |x2 − x1| = 0.75 wherex1 = 0.15 and x2 = 0.90 andfor 15 < t < 40

if x1 < x < x2 and t1 < t < t2 (and I = 0 otherwise).The stimulus triggers a global oscillatory response bycausing an Andronov–Hopf bifurcation, for which thedetails have already been worked out (Lefebvre et al.2009). The linearization and subsequent eigenvalueanalysis of system (1) for spatially homogeneous eigen-modes of the form u j(x, t) = u j(x) + ueλt for u, ∈ R, λ ∈C, yields the characteristic equation

λ + 1 + Reλτ = 0, (2)

for R = 12 [∫

�dyf ′(uon(y)) + ∫

�dyf ′(uof f (y))]. The

case with adaptation is more involved, as we show next.

3 Neural adaptation

While the idealized network architecture of Eq. (1)allows the cellular populations to maintain a steady (oroscillating) level of activity, more realistic network de-scriptions typically include adaptation. As we will see,this local intrinsic component of cellular dynamics playsa crucial role in balancing the excitation-inhibition ratioacross the system. It is therefore an important factor toconsider in the genesis of global oscillations as well aspossible resonance effects. Various biophysical mech-anisms have been shown to cause adaptation, gene-rating distinct effects on the spiking dynamics (Bendaand Herz 2003; Ermentrout et al. 2001; Prescott andSejnowski 2008; Benda et al. 2010). Here we incor-porate an intrinsic linear adaptation that is modeledas a second field, acting locally and subtractively onthe activity field (1). Note that we are not modeling aspecific outward current. Rather, this adaptation fieldcan be seen as a subtractive current that approximatesthe impact of a number of realistic adaptation mech-

anisms on firing rate and resonance properties. Thismodel adaptation field will thus reflect the effect of thislinear component on the evolution of spatio-temporalactivity. Our model now becomes:

(1 + a−1∂t

)uon(x, t) = −A(t − τ) + I(x, t)

− εonwon(x, t)

(1 + a−1∂t

)uof f (x, t) = −A(t − τ) − I(x, t)

− εof f wof f (x, t), (3)

where the adaptation fields won,of f (x, t) obey

(1 + b−1∂t

)won(x, t) = uon(x, t)

(1 + b−1∂t

)wof f (x, t) = uof f (x, t). (4)

Here b is the rate of the adaptation (b−1 is the adap-tation time constant) and εon,of f > 0 corresponds to thegain or amplitude of the adaptation component, whichacts as an inhibitory feedback. System (3) includestwo additive inhibitory components which obey lineardynamics and operate on the time scale b−1, which weassume is identical for ON and OFF cells. Further, theadaptation gain is assumed to be identical for both ONand OFF cells i.e. εon = εof f = ε. The fields won,of f (x, t)are expected to locally inhibit the activity of both ONor OFF populations whenever the ON and OFF activ-ities uon,of f (x, t) increase. The instability threshold forwhich global oscillations can be triggered by static localstimuli will change according to the gain and relativetime scale of this new inhibitory mechanism.

To determine the impact of adaptation, one needsto rework the stability analysis taking into account the


increased dimensionality of the problem. This analysisrelies on the symmetry between ON and OFF cells andis restricted to the choice of identical adaptation gainsand time scales for both sub-populations. The steadystates of the combined systems (3) and (4) are solu-tions of:

(1 + ε)uon(x) = −A(uon, uof f ) + I(x)

(1 + ε)uof f (x) = (1 + ε)uon(x) − 2I(x), (5)

where won = uon(x) and wof f = uof f (x). Adaptationacts as a contracting components, reducing the ampli-tude of the steady states by a factor (1 + εon,of f ) >

0. Considering the spatially homogeneous eigen-modes u j(x, t) = u j(x) + ueλt and w j(x, t) = w j(x) +

weλt, u, w ∈ R, λ ∈ C, one obtains from Eqs. (3) and (4)the Jacobian with delayed components

J(λ)

=

⎛

⎜⎜⎝

−a(1 + Ron(uon)e−λτ ) −aRof f (uof f )e−λτ −aε 0

−aRon(uon)e−λτ −a(1 + Rof f (uof f )e−λτ ) 0 −aε

b 0 −b 0

0 b 0 −b

⎞

⎟⎟⎠

Ron = 12 [∫

�dyf ′(uon(y))] and Rof f = 1

2 [∫�

dyf ′(uof f

(y))]. The characteristic equation follows as

0 = det(J(λ) − λI4), (6)

with I4 being the 4 × 4 identity matrix. The result-ing fourth-order polynomial in λ admits the followingsolutions

λ1,2 = −a + b2

± 12

√b 2 − 2ab + a2 − 4abε,

λ3,4 = −aRe−iwτ + a + b2

±12

√(Rae−λτ )2 + 2a2 Re−λτ − 2ab Re−λτ + a2 − 2ab + b 2 − 4abε, (7)

where the eigenvalues λ3,4 are implicitly determined.The function R can be expressed as

R = Ron(uon) + Rof f (uof f )

= 12

[∫

�

dyf ′(uon(y))] + 1

2

[∫

�

dyf ′ (uof f (y))]

. (8)

The eigenvalues λ1,2 define the stability of Eq. (4) andthus do not depend on the delay τ . Given that a, b , ε >

0, λ1,2 remain bounded to the left of the imaginary axisand subsequently do not contribute to any oscillatoryinstability. They however introduce an additional fre-quency in the system, whenever λ1,2 ∈ C with non-zeroimaginary parts. An input-induced oscillation requiresthe non-linear delayed feedback connections. We maytherefore restrict the analysis to the eigenvalues λ3,4,which depend on the delay τ as well as the parameterR. At the instability threshold, λ3,4 = {iwk|R � wk > 0},and we obtain the same criterion for an Andronov–Hopf bifurcation from both λ3 and λ4, namely

0 = (2iw + aRe−iwτ + a + b

)2 − a2 − 2a2 Re−iwτ

+ 2ab − R2a2e−2iwτ + 2ab Re−iwτ − b 2 + 4abε. (9)

Expanding and separating the real and imaginarycomponents using e−iwτ =cos(wτ)−isin(wτ), we obtain

0 = −w2 + awR sin(wτ) + ab(ε + 1) + ab R cos(wτ)

0 = awR cos(wτ) + (a + b)w − ab R sin(wτ), (10)

Combining these equations, the instability thresholdRc becomes

Rc cos(w(Rc)τ ) = −b 2(ε + 1) + w(Rc)2

b 2 + w(Rc)2 , (11)

for which the frequencies can be shown to be

w(Rc, ε, a, b)

= ±12

∗√

P1(Rc, ε, a, b) ± 2√

P2(Rc, ε, a, b),

with the polynomials

P1(Rc, ε, a, b) = 2(R2

c − 1)

a2 − 2b 2 + 4abε,

P2(Rc, ε, a, b) = b 4 − 4b 3aε − 2a2b 2

+ 2R2ca2b 2 − 4a3bε + 4a3bεR2

c

+ a4 − 2a4 R2c + R4

ca4 − 8b 2a2ε.


Equation (11) defines the instability threshold as afunction of the adaptation gain ε and time scale b−1.The reader might notice that whenever ε, b = 0, onerecovers the eigenvalue problem exposed in Eq. (2), inwhich no adaptation was present.

The problem of determining the overall effect ofcellular adaptation on input-induced oscillations istwofold. First, one must see how ε > 0 and b > 0change the value of the instability threshold Rc inEq. (11) with respect to the case without adaptationi.e. ε = 0, b = 0. Secondly, one must see whether ε > 0reshapes the function R in Eq. (8) by shifting the steadystates. Thus, we must consider the fact that adaptationmight not only change the critical value Rc where anoscillatory response occurs, but also the trajectory inparameter space on which the system reaches stablecyclic solutions.

Figure 3 exposes the effect of increasing gain andadaptation rate constant on the instability threshold Rc

in Eq. (11). For b small, R(b , ε) < Rc. The Andronov–Hopf threshold becomes smaller as the gain increasesε > 0, meaning that oscillatory states require weakerinputs to be reached. As b increases, the oppositeoccurs, and the threshold increases away from the caseε = 0. Cellular adaptation in the electrosensory systemoperates on time scales of roughly 100ms, slower thanthe intrinsic dynamics of the cells, which are on theorder of 10–20 ms (Krahe et al. 2008). As a result, b is

Fig. 3 Oscillatory response threshold Rc as a function of theadaptation gain ε and time constant b−1. Slow adaptation (bsmall) reduces the response threshold marginally, while on fastertime scales, the threshold is increased, reducing the tendency ofthe system to enter oscillatory states of activity in response tostatic stimuli. The adaptation gain ε amplifies both effects. Wenote that for b ≈ 0.8, the gain ε has almost no effect on the valueof Rc. Parameters are a = 1, τ = 2.0, αon,of f = 0.5 and � = [0, 1]

more likely to be smaller than the cellular rate constanta, here fixed to a = 1. For this interval of b values,slow adaptation enhances the prevalence oscillatoryresponses by reducing the value of the bifurcationthreshold.

The effect of local stimulation on the function Rfor ε = 0 has been investigated (Lefebvre et al. 2009),but additional inhibitory components like adaptationinfluence the steady states as well. Indeed, from Eq. (5)and by considering the weak feedback regime i.e.f (uon,of f ) ≈ 0, adaptation reduces the response con-trast by a factor (1 + εon,of f )

−1. Figure 4 shows theresponse amplitude of stimulated cells as a functionof the adaptation gain ε. The input contrast is definedby the activity difference between stimulated and non-stimulated sites for networks with ε = 0 and ε > 0 inthe steady state regime. This definition refers to thedynamics out of any oscillatory regimes and is used tospecify the net impact of inputs on the activity of thesub-units. Once oscillations appear, we use the term“response” instead to qualify the magnitude of theoscillations (see Section 4). As the adaptation gain in-creases, the response amplitude decreases. As a result,the amplitude of the inhibitory feedback for ε > 0 issmaller than for ε = 0; this allows the steady states to

Fig. 4 Input contrast for a static pulse of the form I(x, t) = Ioif x ∈ � and t1 < t < t2 where � corresponds to the input spatialwidth, defined by � = |x1 − x2|. The contrast is the difference inactivity of units inside and outside the pulse. As the adaptationgain increases, the local response decreases. The activity levelshave been chosen such that the feedback connections are weaklyinteracting with the ON and OFF population activities. The solidcurves illustrate the contrast when adaptation is present and asa function of the gain ε for different input amplitudes, while thedashed lines show the response for ε = 0


Fig. 5 Points in (Io, ε) parameter space for which a Andronov–Hopf bifurcation occurs. The input is a stationary pulse as in Fig. 4for a fixed width. (a) Due to the presence of ON and OFF cells,oscillatory responses are observed for both excitatory (Io > 0)and inhibitory (Io < 0) pulses, making the functional R(Io, ε)

symmetric with respect to Io, irrespective of the spatial extentof the stimulus. Because h is small (h = 0.07), as ε increases, theminimal input amplitude causing a Andronov–Hopf bifurcationis smaller, enhancing the tendency of the system towards cyclic

activity, and the interval of values broadens. The vertical darkgray bands correspond to the case without adaptation (ε = 0).(b) For larger feedback thresholds (i.e. h = 0.1), the mean valueof the function R(Io, ε) is smaller for ε > 0. The minimal inputamplitude causing a Andronov–Hopf bifurcation becomes largeras ε increases. Here, the feedback threshold is h = 0.1. Parame-ters are a = 1, τ = 2.0, � = [0, 1]. The pulse width is � = 0.5. Therate constant b was set to 0.8, where the critical value Rc remainsapproximately constant as ε changes (see Fig. 3)

reach higher values. This can be verified numerically bysolving Eq. (5) for both ε > 0 and ε = 0, or analyticallyby considering only the first Taylor expansion termof f .

Adaptation also reduces the magnitude of the feed-back signal sent to the sensory layer in the same waythat it limits the response contrast. Even though thesame amount of feedback connections gets recruited,the amplitude of the return signal is reduced; theamount of inhibition in the system decreases. As such,increasing the adaptation gain does in part increasethe activity of the ON and OFF populations. The be-havior of the system with respect to stimulation is atrade-off between a weaker contrast and inhibition. Asa consequence, this modification of the steady statessignificantly alters the way the system interacts withthe non-linearities of Eq. (1) and thus changes thelocation in parameter space where a bifurcation occurs.To understand this effect, we need to investigate theconsequences of ε > 0 on the shape of the functionR = R(ε) in Eq. (8).

The function R is an integral over the steady statesacross the network, via the derivative of the activationfunction f . It is maximal whenever uon,of f = h, that is,the closer the equilibrium activities are to the thresholdfor firing (and thus for producing feedback activity), thehigher R becomes. It may also be seen as the amount ofnon-linearity in the system. With adaptation, significantvariations of the function R require large input am-plitudes. In the context of a static pulse of width �

and amplitude Io, Fig. 5 illustrates points in (Io, ε) pa-rameter space for which an input-induced Andronov–Hopf bifurcation occurs, i.e. for which R(Io, ε) > Rc. Itillustrates how the function R(Io, ε) behaves accordingto an input amplitude Io and increasing adaptation gainε. A diminished input amplitude implies that largerinput amplitudes are required to bring the system inregions of parameter space where oscillations are stable

Fig. 6 Points in (Io, ε) parameter space for which R(ε) > Rc fora static pulse-shaped stimulus of amplitude Io and various widths.Pulses of smaller widths only generate rhythmic responses whenthe adaptation gain is increased. Regions where the limit cyclesare stable, plotted in shades of gray, are symmetric with respectto the line Io. Parameters are a = 1, τ = 2.0, � = [0, 1], h = 0.07and b = 0.8


(gray regions). But it also implies that once the systemreaches those states, they are more robust and remainstable over a larger range of input amplitudes.

Cellular adaptation diminishes the variability of R.When h is low, more significant contributions are madeby the lateral units to the non-linear connections; thisis even more so as the adaptation gain increases. Thesteady states uon,of f of lateral sites are higher for ε >

0 due to less feedback inhibition, and are typicallyfound closer to the activation threshold h. This resultsin higher values of the function R(Io, ε): the systemremains close to the Andronov–Hopf regime over alarger portion of parameter space because the adapta-tion disposes the lateral activities to maintain a higherdegree of non-linearity. As shown in Fig. 5(a), theminimal amplitude required for global oscillations issmaller for ε > 0 than for ε = 0. For high values of h,

the effect of the adaptation gain on the lateral contri-butions is negligible, since the reduced contrast causesthe function R(Io, ε) to be much smaller when ε > 0than when ε = 0. This is the case depicted in Fig. 5(b).Nevertheless, the broadening of the amplitude intervalsdue to ε makes oscillations more prevalent in a systemthat incorporates adaptation. This supports previousresults on recurrent nets with adaptation, where slowrecurrent components were shown to facilitate the gen-esis of cyclic activity (Crook et al. 1998; Ermentroutet al. 2001; Ly and Ermentrout 2010). We note thatwhenever the input absolute amplitude increases, thesystem first enters the Andronov–Hopf regime whereglobal oscillations are stable; but if the input amplitudebecomes too high, the oscillations disappear via a re-verse Andronov–Hopf bifurcation. This is so becausethe sub-unit activities are taken to values much higher

Fig. 7 Response amplitude of ON cells to a spatially localizedpulse sinusoidally modulated in time i.e I(x) = Iosin(wot) forx ∈ �. The response is defined as the difference between themaximal amplitude of the activity reached by the solutions, andthe activity level prior to any stimulus. For each trial, the adap-tation time scale is increased, from slow to fast: 1. b = 0.2; 2.b = 0.5; 3. b = 0.8; 4. b = 1.1. (a) With non-delayed feedback,the response is constant over the range of input frequencies. Thesystem responds maximally to the adaptation intrinsic frequency,while the inhibitory feedback keeps the responses weak. Onedoes not see the Hopf frequency because of the zero delay. (b)Without feedback, the system demonstrates strong resonance

near the adaptation frequencies. The response decreases in am-plitude as the adaptation becomes faster. (c) When delayedfeedback is considered, dominant responses are seen near theHopf frequency, where the change in adaptation time-scale doesnot seem to significantly shift the resonances, although the dy-namics still become high pass. The delay chosen is τ = 1.5. d)Without adaptation (i.e. ε = 0), the resonance occurs at the Hopffrequency, and the response amplitude is generally larger (evenfor wo = 0.0). Parameters are a = 1, � = [0, 1], h = 0.1, Io = 0.2and � = 0.4. The adaptation gain was set in panel (a)–(c) toε = 0.6


than the threshold h, where the feedback A behaveslinearly. The activity of ON and OFF cells is theninhibited sufficiently by the feedback to destroy theoscillations.

Our analytic work assumes that the adaptation rateand gains are equal between ON and OFF populations.However, it has been shown that in the electrosensorysystem these parameters vary between ON and OFFpopulations, and also across the various electrosensoryspatial maps (Krahe et al. 2008; Mehaffey et al. 2008).In fact, this motivates varying the adaptation ratesbelow in Fig. 7. If the symmetry restriction is relaxedand two distinct adaptation rates b on and b of f are in-troduced, the Andronov–Hopf regions of Fig. 5 becomenon-symmetric with respect to the vertical line Io = 0,meaning that the system does not respond evenly toexcitatory and inhibitory inputs anymore (not shown).In fact, the sub-population with the slowest adaptationdominates: whenever b of f < b on, a wider range of in-hibitory input amplitudes triggers oscillatory activity,i.e. the system becomes more sensitive to inhibitoryinputs because OFF cell adaptation is slower. If b on <

b of f , the opposite occurs. This effect is amplified withthe magnitude of |b of f − b on|.

An interesting consequence of these effects is thatthe system may now respond to pulses of smaller spatialextent, in a regime where h is smaller. As the meanvalue of R is larger for ε > 0 in this case, input causingsmall fluctuations in the function R now becomes acandidate to cause a bifurcation, via either amplitudeor width changes. This would not be the case if adap-tation were not present, because the mean value ofR would be much smaller. This effect is caused by asmaller amount of inhibition fed back to the sensorylayer by the recurrent connections, as the units adaptas well to this steady return current. Figure 6 showshow the regions in (Io, ε) space changes as a functionof input width. As the width � diminishes from 0.50to 0.35, regions of stable oscillatory activity retract to-wards higher adaptation gains, meaning that adaptationis necessary for those smaller inputs to trigger stableoscillatory solutions. We note that this behavior occursbecause of the choice of a small feedback threshold h.

In Lefebvre et al. (2010), it was shown that if anadditional non-delayed inhibitory feedback componentis added to Eq. (1), the Andronov–Hopf thresholdincreases, thus reducing the tendency of the system toundergo oscillatory behavior with respect to spatiallylocalized inputs. In many aspects, an adaptation currentlike the one considered here plays the same role asan extra instantaneous inhibitory feedback. As Fig. 5shows, the minimal input amplitude required to gen-erate global oscillations increases, which corroborates

the results presented in Lefebvre et al. (2010) for thisparticular choice of large threshold values. This seemsto contradict the results of Fig. 6, but this is not so.While the main effect of adaptation is to move equi-libria in phase space, the non-delayed feedback com-ponents considered in Lefebvre et al. (2010) changedthe bifurcation threshold considerably, which is not thecase here. The oscillations triggered by small pulses asin Fig. 5 are the result of a special choice of h whichbrings the system close to the Andronov–Hopf regime.

4 Time-varying inputs

The results presented in the foregoing analysis arebased on changes of stability of steady states. How-ever, one of the major role of cellular adaptation isto filter time-varying signals. Fast inputs are likely tosignificantly increase the cellular activity until the adap-tive forces activate. This adaptive component typicallyacts on the system after the initial, transient response ofthe cells. As a result, apart from its effect on equilibria,adaptation influences the responses of the system withrespect to fast varying signals. As a final part of ouroverview of adaptation, we outline how the integrationof time-dependent inputs varies according to the adap-tation parameters ε and b . Real sensory signals usuallydemonstrate a high degree of noise, which can includesignificant high frequencies. However, studying time-periodic signals may highlight the frequency tuningproperties of our model.

The behavior of the solutions with respect to time-dependent inputs might help to understand the roleof adaptation and its timing in transient dynamics. InFig. 7, we plot the response amplitude of the ON cellsto a time-periodic signal of frequency wo for variousadaptation time scales. It is evident that as the adap-tation rate constant b increases, the system filters outlower frequencies, thus acting more like a high-passfilter. When no adaptation is present (Fig. 7(d)), thesystem’s maximal response occurs precisely at the Hopffrequency, as the input resonates with the intrinsicoscillations of the system. If the feedback connectionsare removed (Fig. 7(b)), the system becomes fully linearand input frequencies generating maximal responsesare entirely determined by the amplitude of the solu-tions of Eq. (3) which are function of a,b , wo and ε.In particular, as adaptation becomes faster (i.e. whenb increases), the response peak is also shifted towardshigher input frequencies. A similar behavior occurs inthe case with non-delayed feedback (i.e. setting τ =0), while the system responds maximally at the same


frequencies as in Fig. 7(b), although the amplitudes ofthe oscillations are significantly smaller.

When both delayed feedback and adaptation(Fig. 7(c)) are considered, the system responds maxi-mally at an input frequency near the Hopf frequency,which corresponds to a mixture of the cases seen inFig. 7(b) and (d). In this case, the maximal responsefrequency shift is much less significant, as the systemappears to remain closer to the Hopf frequency as bincreases. Increasing the adaptation gain ε amplifiesthis effect (not shown).

The qualitative shape of the curves shown in Fig. 7has also been obtained using a noisy Integrate-and-Firemodel (LIF) that possesses both global feedback andadaptation, where the architecture is the same as inEqs. (3) and (4). The goal here is not to perform a thor-ough comparison of the neural field model dynamics tothose of the LIF model—this will be left as part of afuture study that will consider more biophysically real-istic models of the electrosensory lateral line pyramidalcells, including adaptation, feedback and SK channels.Rather the goal is to show that an LIF description canreproduce the main qualitative features seen in our

neural field model. The evolution of the membranepotential of the jth LIF neuron, for j = 1...N obeys

dvonj (t)

dt= − von

j + g∑

ti

η(ti − τ) − εwonj (t)

+ μ + ξ(t) + I( j, t)

dvof fj (t)

dt= − v

of fj + g

∑

ti

η(ti − τ) − εwof fj (t)

+ μ + ξ(t) − I( j, t)

b−1dwon

j (t)

dt= − won

j (t) + vonj (t)

b−1dw

of fj (t)

dt= − w

of fj (t) + v

of fj (t), (12)

with Gaussian white noise ξ(t) of intensity D, i.e. theautocorrelation is 〈ξ(t)ξ(t′)〉 = 2Dδ(t − t′). The feed-back gain is denoted by g. The network contains NON cells and N OFF cells, which receive inputs of theform I( j, t). The individual spike times of neurons aredenoted by ti, and the bias current by μ. The synaptic

Fig. 8 Mean firing ratefluctuations of stimulated ONcells in a noisyIntegrate-and-Fire net as inEq. (12). The input is aspatially localized pulsesinusoidally modulated intime i.e I(x) = Iosin(wot) forx ∈ �. The plotted firing ratescorrespond to deviationsfrom the non-stimulatedstate. As in Fig. 7, theadaptation time scale isincreased in each panel:1. b = 0.2; 2. b = 0.5; 3.b = 0.8; 4.b = 1.1. The circuitfeatures are also changed ineach panel: (a) withnon-delayed feedback i.e.τ = 0.0; (b) without feedback;(c) with delayed feedback i.e.τ = 1.5; and (d) withoutadaptation (i.e. ε = 0). Ineach case, the behaviorobserved with the neural fieldmodel is reproducedqualitatively. Parameters areIo = 1.0, � = [0, 1], h = 1.0,g = −0.2, μ = 1.05 and� = 0.4. The noise has anamplitude D = 1.0. Theadaptation gain was set inpanels (a)–(c) to ε = 1.5


response function η is an in Eq. (1). The membranetime contsant was set to 1, while the refractory periodis set to τm = 1. Here as well, b−1 stands for the adap-tation rate and ε is the adaptation gain.

Numerical results on resonance properties for theLIF with adaptation are shown in Fig. 8. The parame-ters of the LIF model with ON and OFF cells have beenscaled to fit the neural field description, as in Lefebvreet al. (2009). We see that the LIF model qualitativelyreproduces the behavior of the resonance when theequivalent parameters are varied. This is particularlythe case for the adaptation rate constant. Note that, inthe LIF model, the increase in the feedback delay onlychanges the resonance slightly in comparison to theneural field model. This is likely due to the fact that theLIF includes a feedback kernel which convolves everyspike emitted with a smooth function. This kernel al-ready implements an “equivalent” delay, which alreadycauses a resonance similar to the neural field with delay.Nevertheless, the response curves behave as in Fig. 7 toan increase of the adaptation time scale b , where theprocesses become more high pass.

5 Discussion

In this article, we studied the effects of cellular adapta-tion on the genesis of oscillatory responses to spatiallylocalized static or time-periodic pulses of varying am-plitudes in a recurrent network of ON and OFF cells.Based on previous results, we performed the bifurca-tion analysis for an Andronov–Hopf bifurcation causedby an external stimulus in an adaptive system, andshowed that both the time scale b−1 and gain ε of theadaptation term modified the instability threshold. Theadaptation was shown to decrease the input contrastand the effect of negative feedback, resulting in anincrease of steady state activities. It was further shownthat for weak values of the adaptation rate and highvalues of the gain, adaptation enhances the genesis ofoscillatory responses to stationary pulses.

Specifically, the results of Section 3 demonstrate thatwhenever the adaptation rate b is chosen to be smallenough (resulting in slow adaptation dynamics), cyclicsolutions are found to be more robust compared toa system that does not adapt at all. In this regime,adaptation causes the Andronov–Hopf threshold to besmaller. This was shown by performing a bifurcationanalysis near the fixed points of our model for the casewhere the adaptation timing and gain are identical forboth ON and OFF cells, and by investigating the effectof these on the bifurcation point. A complementary

effect of adaptation may be elucidated by looking athow the activity equilibria behave in the presence ofthe terms won and wof f . Indeed, the adaptation gainaffects the input contrast by decreasing the responseamplitude of the stimulated units. A direct consequenceof this is a wider interval of input amplitudes generatingoscillations, making the system more sensitive to inputsof smaller widths. These results point toward a preva-lence of oscillatory states generated by sensory stimuli.This result is also consistent with the findings on neuraloscillators which show that adaptation enhances thesynchronization properties of networks (Crook et al.1998; Ermentrout et al. 2001), where oscillatory solu-tions become stable in the presence of adaptation butaren’t if feedback alone governs the dynamics (Ly andErmentrout 2010; Kilpatrick and Bressloff 2010).

However, this conclusion holds only in a regimewhere the dynamics of won,of f are slow. In this case, theadaptation components may be seen as an additionalsource of slow inhibitory feedback (albeit instanta-neous rather than delayed), reinforcing the presence ofcyclic activity. If the adaptation becomes fast comparedto the intrinsic dynamics of the units, the oppositeoccurs, and global oscillations become more difficultto obtain. This supports the results on a similar modelwithout adaptation, where it was shown that additionalnon-delayed (and fast) inhibitory feedback componentspushed back (i.e. raised) the Andronov–Hopf threshold(Lefebvre et al. 2010).

Adaptation is also involved in the integration oftemporal signals. It is thus important to see how thiscombines with the resonant properties of ON/OFF nets.In Section 4, resonance curves were plotted, in caseswith and without feedback. The adaptation componentintroduces additional resonances in the system. Thisis apparent by looking at the resonance curves whenno feedback is present, or when the feedback delayis chosen to be zero. The Hopf frequency, located atinput frequencies where the response amplitude is max-imal, depends on the choice of adaptation time scale.The presence of these new frequencies, brought up bytwo additional complex eigenvalues in Eq. (7), seemsto corroborate previous results on frequency tuningregulation properties due to adaptation (Benda andHerz 2003; Mehaffey et al. 2008). Our results are alsoreproduced by numerical simulations on a network ofnoisy LIF cells with an equivalent circuitry.

In the perspective of extending the current model,the use of topographic feedback connections wouldfurther increase the connection of our work to theelectrosensory system. Although an inhibitory spatiallydiffuse feedback connection does exist between highernuclei and the sensory layer, other spatially organized


feedback connections branch to the pyramidal cellswith glutamatergic as well as gabaergic connections.The topographic feedback in fact brings in a true spa-tial dimension to the all-to-all connected model. Thesefeedback connections would greatly influence the sta-bility of the activity distributions, especially regardingthe presence of spatio-temporal stimuli. Topographicfeedback, along with adaptation currents, has beenshown to influence the stability of activity patterns andthe propagation of oscillations (Folias and Bressloff2005; Kilpatrick and Bressloff 2010). We thus expectthat this will also be the case in our model. It would alsobe of interest to test whether the dynamics illustratedhere differ if ON cells feedback more predominantly toON cells, and the same for OFF cells. This could leadto predictions about this feedback connectivity.

Further, experimental studies on the weakly electricfish have shown that adaptation time scales vary acrossthe different sensory maps into which the ELL is di-vided (Krahe et al. 2008). Likewise, the receptive fieldsize of pyramidal cells increases going from central tolateral positions. In this context, it would be interestingto determine how frequency tuning properties of thesemaps relate to the adaptation time scales when ON andOFF cells are present, and how this tuning depends onreceptive field size and topographic feedback. Finally,ON and OFF cells may have different firing rates in theabsence of input (I(x, t) = 0). This activity difference,when sufficiently strong, can influence the dynamicsof recurrent ON/OFF in the absence of adaptation(Lefebvre et al. 2009). The role that it may play in thefrequency tuning of cells and oscillation susceptibilityof the network with adaptation thus remains to beinvestigated.

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