{"title": "Spike Frequency Adaptation Implements Anticipative Tracking in Continuous Attractor Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 505, "page_last": 513, "abstract": "To extract motion information, the brain needs to compensate for time delays that are ubiquitous in neural signal transmission and processing. Here we propose a simple yet effective mechanism to implement anticipative tracking in neural systems. The proposed mechanism utilizes the property of spike-frequency adaptation (SFA), a feature widely observed in neuronal responses. We employ continuous attractor neural networks (CANNs) as the model to describe the tracking behaviors in neural systems. Incorporating SFA, a CANN exhibits intrinsic mobility, manifested by the ability of the CANN to hold self-sustained travelling waves. In tracking a moving stimulus, the interplay between the external drive and the intrinsic mobility of the network determines the tracking performance. Interestingly, we find that the regime of anticipation effectively coincides with the regime where the intrinsic speed of the travelling wave exceeds that of the external drive. Depending on the SFA amplitudes, the network can achieve either perfect tracking, with zero-lag to the input, or perfect anticipative tracking, with a constant leading time to the input. Our model successfully reproduces experimentally observed anticipative tracking behaviors, and sheds light on our understanding of how the brain processes motion information in a timely manner.", "full_text": "Spike Frequency Adaptation Implements Anticipative\nTracking in Continuous Attractor Neural Networks\n\nYuanyuan Mi\n\nState Key Laboratory of Cognitive Neuroscience & Learning,\n\nBeijing Normal University,Beijing 100875,China\n\nmiyuanyuan0102@bnu.edu.cn\n\nDepartment of Physics, The Hong Kong University of Science and Technology, Hong Kong\n\nC. C. Alan Fung, K. Y. Michael Wong\n\nphccfung@ust.hk, phkywong@ust.hk\n\nState Key Laboratory of Cognitive Neuroscience & Learning, IDG/McGovern Institute for\n\nBrain Research, Beijing Normal University, Beijing 100875, China\n\nSi Wu\n\nwusi@bnu.edu.cn\n\nAbstract\n\nTo extract motion information, the brain needs to compensate for time delays that\nare ubiquitous in neural signal transmission and processing. Here we propose a\nsimple yet effective mechanism to implement anticipative tracking in neural sys-\ntems. The proposed mechanism utilizes the property of spike-frequency adapta-\ntion (SFA), a feature widely observed in neuronal responses. We employ con-\ntinuous attractor neural networks (CANNs) as the model to describe the tracking\nbehaviors in neural systems. Incorporating SFA, a CANN exhibits intrinsic mo-\nbility, manifested by the ability of the CANN to support self-sustained travelling\nwaves. In tracking a moving stimulus, the interplay between the external drive\nand the intrinsic mobility of the network determines the tracking performance. In-\nterestingly, we \ufb01nd that the regime of anticipation effectively coincides with the\nregime where the intrinsic speed of the travelling wave exceeds that of the external\ndrive. Depending on the SFA amplitudes, the network can achieve either perfect\ntracking, with zero-lag to the input, or perfect anticipative tracking, with a con-\nstant leading time to the input. Our model successfully reproduces experimentally\nobserved anticipative tracking behaviors, and sheds light on our understanding of\nhow the brain processes motion information in a timely manner.\n\n1 Introduction\n\nOver the past decades, our knowledge of how neural systems process static information has ad-\nvanced considerably, as is well documented by the receptive \ufb01eld properties of neurons. The equally\nimportant issue of how neural systems process motion information remains much less understood.\nA main challenge in processing motion information is to compensate for time delays that are perva-\nsive in neural systems. For instance, visual signal transmitting from the retina to the primary visual\ncortex takes about 50-80 ms [1], and the time constant for single neurons responding to synaptic\ninput is of the order 10-20 ms [2]. If these delays are not compensated properly, our perception of a\nfast moving object will lag behind its true position in the external world signi\ufb01cantly, impairing our\nvision and motor control.\n\n1\n\n\fA straightforward way to compensate for time delays is to anticipate the future position of a moving\nobject, covering the distance the object will travel through during the delay period. Experimental\ndata has suggested that our brain does employ such a strategy. For instance, it was found that in spa-\ntial navigation, the internal head-direction encoded by anterior dorsal thalamic nuclei (ADN) cells\nin a rodent was leading the instant position of the rodent\u2019s head by \u223c 25 ms [3, 4, 5], i.e., it was the\ndirection the rodent\u2019s head would turn into \u223c 25 ms later. Anticipation also justi\ufb01es the well-known\n\ufb02ash-lag phenomenon [6], that is, the perception that a moving object leads a \ufb02ash, although they\ncoincide with each other at the same physical location. The reason is due to the anticipation of our\nbrain for the future position of the continuously moving object, in contrast to the lack of anticipa-\ntion for intermittent \ufb02ashes. Although it is clear that the brain do have anticipative response to the\nanimal\u2019s head direction, it remains unclear how neural systems implement appropriate anticipations\nagainst various forms of delays.\nDepending on the available information, the brain may employ different strategies to implement\nanticipations.\nIn the case of self-generated motion, the brain may use an efference copy of the\nmotor command responsible for the motion to predict the motion consequence in advance [7]; and\nin the case when there is an external visual cue, such as the speed of a moving object, the neural\nsystem may dynamically select a transmission route which sends the object information directly to\nthe future cortical location during the delay [8]. These two strategies work well in their own feasible\nconditions, but they may not compensate for all kinds of neural delays, especially when the internal\nmotor command and visual cues are not available. Notably, it was found that when a rodent was\nmoving passively, i.e., a situation where no internal motor command is available, the head-direction\nencoded by ADN cells was still leading the actual position of the rodent\u2019s head by around \u223c 50ms,\neven larger than that in a free-moving condition [5]. Thus, extra anticipation strategies may exist in\nneural systems.\nHere, we propose a novel mechanism to generate anticipative responses when a neural system is\ntracking a moving stimulus. This strategy does not depend on the motor command information\nnor external visual cues, but rather relies on the intrinsic property of neurons, i.e., spike-frequency\nadaptation (SFA). SFA is a dynamical feature commonly observed in the activities of neurons when\nthey have experienced prolonged \ufb01ring. It may be generated by a number of mechanisms [10]. In\none mechanism, neural \ufb01ring elevates the intracellular calcium level of a neuron, which induces an\ninward potassium current and subsequently hyperpolarizes the neuronal membrane potential [11].\nIn other words, strong neuronal response induces a negative feedback to counterbalance itself. In the\npresent study, we use continuous attractor neural networks (CANNs) to model the tracking behaviors\nin neural systems. It was known that SFA can give rise to travelling waves in CANNs [12] analogous\nto the effects of asymmetric neuronal interactions; here we will show that its interplay with external\nmoving stimuli determines the tracking performance of the network. Interestingly, we \ufb01nd that when\nthe intrinsic speed of the network is greater than that of the external drive, anticipative tracking\noccurs for suf\ufb01ciently weak stimuli; and different SFA amplitude results in different anticipative\ntimes.\n\n2 The Model\n\n2.1 Continuous attractor neural networks\n\nWe employ CANNs as the model to investigate the tracking behaviors in neural systems. CANNs\nhave been successfully applied to describe the encoding of continuous stimuli in neural systems,\nincluding orientation [13], head-direction [14], moving direction [15] and self location [16]. Re-\ncent experimental data strongly indicated that CANNs capture some fundamental features of neural\ninformation representation [17].\nConsider a one-dimensional continuous stimulus x encoded by an ensemble of neurons (Fig. 1).\nThe value of x is in the range of (\u2212\u03c0, \u03c0] with the periodic condition imposed. Denote U (x, t) as the\nsynaptic input at time t of the neurons whose preferred stimulus is x, and r(x, t) the corresponding\n\ufb01ring rate. The dynamics of U (x, t) is determined by the recurrent input from other neurons, its own\nrelaxation and the external input I ext(x, t), which is written as\n\n\u222b\n\n\u03c4\n\ndU (x, t)\n\ndt\n\n= \u2212U (x, t) + \u03c1\n\nJ(x, x\n\n\u2032\n\n\u2032\n)r(x\n\n, t)dx\n\n\u2032\n\n+ I ext(x, t),\n\n(1)\n\nx\u2032\n\n2\n\n\f\u2032\n\nFigure 1: A CANN encodes a continuous stimulus, e.g., head-direction. Neurons are aligned in\nthe network according to their preferred stimuli. The neuronal interaction J(x, x\n) is translation-\ninvariant in the space of stimulus values. The network is able to track a moving stimulus, but\nthe response bump U (x, t) is always lagging behind the external input I ext(x, t) due to the neural\nresponse delay.\nwhere \u03c4 is the synaptic time constant, typically of the order 2 \u223c 5 ms, \u03c1 is the neural density and\n\u2032 to x, where the Gaussian\nJ(x, x\nwidth a controls the neuronal interaction range. We will consider a \u226a \u03c0. Under this condition, the\nneuronal responses are localized and we can effectively treat \u2212\u221e < x < \u221e in the following\nanalysis.\nThe nonlinear relationship between r(x, t) and U (x, t) is given by\n\n[\u2212(x \u2212 x\n\nis the neural interaction from x\n\n) = J0\u221a\n\n)2/(2a2)\n\n]\n\nexp\n\n2(cid:25)a\n\n\u2032\n\n\u2032\n\n\u222b\nU (x, t)2\nx\u2032 U (x\u2032, t)2dx\u2032 ,\n\nr(x, t) =\n\n1 + k\u03c1\n\n(2)\n\nwhere the divisive normalization could be realized by shunting inhibition. r(x, t) \ufb01rst increases\nwith U (x, t) and then saturates gradually when the total network activity is suf\ufb01ciently large. The\nparameter k controls the strength of divisive normalization. This choice of global normalization can\nsimplify our analysis and should not alter our main conclusion if localized inhibition is considered.\nIt can be checked that when I ext = 0, the network supports a continuous family of Gaussian-shaped\nstationary states, called bumps, which are,\n\n[\n\n]\n\n[\n\n]\n\nU (x) = U0exp\n\n\u221a\n\u221a\n1 \u2212 8\n\u221a\n2\u03c0ak/(\u03c1J 2\n2\u03c0a).\n\n\u2212 (x \u2212 z)2\n\n\u2212 (x \u2212 z)2\n\u221a\n2U0/(\u03c1J0) and U0 =\n\u221a\n2\u03c0ak\u03c1). The bumps are stable for 0 < k < kc, with kc =\n0 )]/(2\n\nr(x) = r0exp\n\nr0 =\n\n\u2200z\n\n2a2\n\n4a2\n\n(3)\n\n,\n\n,\n\n0 /(8\n\n[1 +\n\u03c1J 2\nThe bump states of a CANN form a sub-manifold in the state space of the network, on which the\nnetwork is neutrally stable. This property enables a CANN to track a moving stimulus smoothly,\nprovided that the stimulus speed is not too large [18]. However, during the tracking, the network\nbump is always lagging behind the instant position of the moving stimulus due to the delay in\nneuronal responses (Fig. 1).\n\nwhere the peak position of the bump z is a free parameter.\n\n2.2 CANNs with the asymmetrical neuronal interaction\n\nIt is instructive to look at the dynamical properties of a CANN when the asymmetrical neuronal\ninteraction is included. In an in\ufb02uential study [14], Zhang proposed an idea of adding asymmetrical\ninteractions between neurons in a CANN, such that the network can support travelling waves, i.e.,\nspontaneously moving bumps. The modi\ufb01ed model well describes the experimental \ufb01nding that in\ntracking the rotation of a rodent, the internal representation of head-direction constructed by ADN\ncells also rotates and the bump of neural population activity remains largely invariant in the rotating\nframe.\nBy including the asymmetrical neuronal interaction, the CANN model presented above also supports\na travelling wave state. The new neuronal recurrent interaction is written as\n\n]\n\n]\n\n[\n\n[\n\n~J(x, x\n\n\u2032\n\n) =\n\nJ0\u221a\n2\u03c0a\n\nexp\n\n\u2212 (x \u2212 x\n\u2032\n\n)2\n\n2a2\n\n+ \u03b3\u03c4\n\nJ0\u221a\n2\u03c0a3\n\n(x \u2212 x\n\n\u2032\n\n) exp\n\n3\n\n\u2212 (x \u2212 x\n\u2032\n\n)2\n\n2a2\n\n,\n\n(4)\n\n(cid:54)(cid:9)(cid:89)(cid:13)(cid:85)(cid:10)(cid:42)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:89)(cid:13)(cid:85)(cid:10)(cid:70)(cid:89)(cid:85)J(x,x\u2019)(cid:87)(cid:70)(cid:89)(cid:85)(cid:69)(cid:70)(cid:77)(cid:66)(cid:90)\f{\n}\n\u2212 [x \u2212 (z + vt)]2 /(4a2)\n\nwhere \u03b3 is a constant controlling the strength of asymmetrical interaction.\nIt is straightforward to check that the network supports the following traveling wave solution,\n\u2212 [x \u2212 (z + vt)]2 /(2a2)\n,\nU (x, t) = U0 exp\nwhere v is the speed of the travelling wave, and v = \u03b3, i.e., the asymmetrical interaction strength\ndetermines the speed of the travelling wave (see Supplementary Information).\n\nr(x, t) = r0 exp\n\n}\n\n{\n\n,\n\n2.3 CANNs with SFA\n\nThe aim of the present study is to explore the effect of SFA on the tracking behaviors of a CANN.\nIncorporating SFA, the dynamics of a CANN is written as\n\n\u03c4\n\ndU (x, t)\n\ndt\n\n= \u2212U (x, t) + \u03c1\n\n\u2032\n\n\u2032\n)r(x\n\n, t)dx\n\n\u2032 \u2212 V (x, t) + I ext(x, t),\n\nJ(x, x\n\n(5)\n\nx\u2032\n\n\u222b\n\nwhere the synaptic current V (x, t) represents the effect of SFA, whose dynamics is given by [12]\n\ndV (x, t)\n\n= \u2212V (x, t) + mU (x, t),\n\ndt\n\n\u03c4v\n\n(6)\nwhere \u03c4v is the time constant of SFA, typically of the order 40 \u223c 120 ms. The parameter m controls\nthe SFA amplitude. Eq. (6) gives rise to V (x, t) = m\n/\u03c4v, that\nis, V (x, t) is the integration of the neuronal synaptic input (and hence the neuronal activity) over\nan effective period of \u03c4v. The negative value of V (x, t) is subsequently fed back to the neuron to\nsuppress its response (Fig. 2A). The higher the neuronal activity level is, the larger the negative\nfeedback will be. The time constant \u03c4v \u226b \u03c4 indicates that SFA is slow compared to neural \ufb01ring.\n\n\u222b\nt\u2212\u221e exp [\u2212(t \u2212 t\n\n)/\u03c4v] U (x, t\n\n\u2032\n)dt\n\n\u2032\n\n\u2032\n\n\u2211\nFigure 2: A. The inputs to a single neuron in a CANN with SFA, which include a recurrent input\nj Jijrj from other neurons, an external input I ext(x, t) containing the stimulus information, and a\nnegative feedback current \u2212V (x, t) representing the SFA effect. The feedback of SFA is effectively\ndelayed by time \u03c4v. B. The intrinsic speed of the network vint (in units of 1/\u03c4) increases with the\nSFA amplitude m. The network starts to have a travelling wave state at m = \u03c4 /\u03c4v. The parameters\nare: \u03c4 = 1, \u03c4v = 60, a = 0.5. Obtained by Eq. (13).\n\n3 Travelling Wave in a CANN with SFA\n\nWe \ufb01nd that SFA has the same effect as the asymmetrical neuronal interaction on retaining travelling\nwaves in a CANN. The underlying mechanism is intuitively understandable. Suppose that a bump\nemerges at an arbitrary position in the network. Due to SFA, those neurons which are most active\nreceive the strongest negative feedback, and their activities will be suppressed accordingly. Under\nthe competition (mediated by recurrent connections and divisive normalization) from the neighbor-\ning neurons which are less affected by SFA, the bump tends to shift to the neighborhood; and at the\nnew location, SFA starts to destabilize neuronal responses again. Consequently, the bump will keep\nmoving in the network like a travelling wave.\nThe condition for the network to support a travelling wave state can be theoretically analyzed. In\nsimulations, we observe that in a traveling wave state, the pro\ufb01les of U (x, t) and V (x, t) have ap-\nproximately a Gaussian shape, if m is small enough. We therefore consider the following Gaussian\n\n4\n\n(cid:34)(cid:35)-V(x,t) : SFA v\u03c4(,)extIxtU(x,t)ijjjJr\u221100.020.040.060.08\u22120.00500.0050.010.0150.020.025\u03c4 / \u03c4vmvint\fansatz for the travelling wave state,\n\nU (x, t) = Au exp\n\nr(x, t) = Ar exp\n\nV (x, t) = Av exp\n\n}\n}\n\n{\n\u2212 [x \u2212 z(t)]2\n{\n\u2212 [x \u2212 z(t)]2\n{\n\u2212 [x \u2212 (z(t) \u2212 d)]2\n\n2a2\n\n4a2\n\n,\n\n,\n\n4a2\n\n}\n\n,\n\n(7)\n\n(8)\n\n(9)\n\nwhere dz(t)/dt is the speed of the travelling wave and d is the separation between U (x, t) and\nV (x, t). Without loss of generality, we assume that the bump moves from left to right, i.e.,\ndz(t)/dt > 0. Since V (x, t) lags behind U (x, t) due to slow SFA, d > 0 normally holds.\nTo solve the network dynamics, we utilize an important property of CANNs, that is, the dynamics of\na CANN are dominated by a few motion modes corresponding to different distortions in the shape\nof a bump [18]. We can project the network dynamics onto these dominating modes and simplify the\nnetwork dynamics signi\ufb01cantly. The \ufb01rst two dominating motion modes used in the present study\ncorrespond to the distortions in the height and position of the Gaussian bump, which are given by\n\u03d50(x|z) = exp\n. By projecting\na function f (x) onto a mode \u03d5n(x), we mean computing\nApplying the projection method, we solve the network dynamics and obtain the travelling wave state.\nThe speed of the travelling wave and the bumps\u2019 separation are calculated to be (see Supplementary\nInformation)\n\n[\u2212(x \u2212 z)2/(4a2)\n]\nand \u03d51(x|z) = (x \u2212 z) exp\n\u221a\n\n[\u2212(x \u2212 z)2/(4a2)\n\u221a\n\nx f (x)\u03d5n(x)dx/\n\nx \u03d5n(x)2dx.\n\n\u221a\n\n\u222b\n\n\u222b\n\n]\n\n\u03c4\n\nm\u03c4v\n\n,\n\nvint \u2261 dz(t)\n\ndt\n\n=\n\n2a\n\u03c4v\n\n\u2212\n\nm\u03c4v\n\n\u03c4\n\nm\u03c4v\n\n\u03c4\n\n.\n\n(10)\n\n\u221a\n1 \u2212\n\nd = 2a\n\nThe speed of the travelling wave re\ufb02ects the intrinsic mobility of the network, and its value is fully\ndetermined by the network parameters (see Eq. (10)). Hereafter, we call it the intrinsic speed of the\nnetwork, referred to as vint. vint increases with the SFA amplitude m (Fig. 2B). The larger the value\nof vint, the higher the mobility of the network.\nFrom the above equations, we see that the condition for the network to support a travelling wave\nstate is m > \u03c4 /\u03c4v. We note that SFA effects can reduce the \ufb01ring rate of neurons signi\ufb01cantly [11].\nSince the ratio \u03c4 /\u03c4v is small, it is expected that this condition can be realistically ful\ufb01lled.\n\n3.1 Analogy to the asymmetrical neuronal interaction\n\nBoth SFA and the asymmetrical neuronal interaction have the same capacity of generating a travel-\nling wave in a CANN. We compare their dynamics to unveil the underlying cause.\nConsider that the network state is given by Eq. (8). The contribution of the asymmetrical neuronal\ninteraction can be written as (substituting the asymmetrical component in Eq. (4) into the second\nterm on the right-hand side of Eq. (1)),\n\n\u221a\nJ0\u03c1\u03b3\u03c4 r0\n2\u03c0a3\n\n(x \u2212 x\n\u2032\n\n)e\n\nx\u2032\n\n\u2032\n\u2212 (x\u2212x\n\n)2\n\n2a2 e\n\n\u2212 (x\n\n\u2032\u2212z)2\n\u2032\n2a2 dx\n\n=\n\n\u03c1J0r0\u03b3\u03c4 (x \u2212 z)\n\n\u221a\n2a2\n2\n\n\u2212 (x\u2212z)2\n4a2\n\ne\n\n.\n\n(11)\n\n\u222b\n\nIn a CANN with SFA, when the separation d is suf\ufb01ciently small, the synaptical current induced by\nSFA can be approximately expressed as (the 1st-order Taylor expansion; see Eq. (9)),\n\n\u2212V (x, t) \u2248 \u2212Av exp\n\n\u2212 (x \u2212 z)2\n\n4a2\n\nx \u2212 z\n2a2 exp\n\n\u2212 (x \u2212 z)2\n\n4a2\n\n+ dAv\n\n,\n\n(12)\n\n[\n\n]\n\n[\n\n]\n\nwhich consists of two terms: the \ufb01rst one has the same form as U (x, t) and the second one has the\nsame form as the contribution of the asymmetrical interaction (compared to Eq. (11)). Thus, SFA\nhas the similar effect as the asymmetrical neuronal interaction on the network dynamics.\nThe notion of the asymmetrical neuronal interaction is appealing for retaining a travelling wave in a\nCANN, but its biological basis has not been properly justi\ufb01ed. Here, we show that SFA may provide\n\n5\n\n\fan effective way to realize the effect of the asymmetrical neuronal interaction without recruiting the\nhard-wired asymmetrical synapses between neurons. Furthermore, SFA can implement travelling\nwaves in either direction, whereas, the hard-wired asymmetrical neuronal connections can only\nsupport a travelling wave in one direction along the orientation of the asymmetry. Consequently, a\nCANN with the asymmetric coupling can only anticipatively track moving objects in one direction.\n\n4 Tracking Behaviors of a CANN with SFA\n\nof\n\nloss\n\ngenerality, we\n\n{\n}\n\u2212 [x \u2212 z0(t)]2 /(4a2)\n\nSFA induces intrinsic mobility of the bump states of a CANN, manifested by the ability of the net-\nwork to support self-sustained travelling waves. When the network receives an external input from a\nmoving stimulus, the tracking behavior of the network will be determined by two competing factors:\nthe intrinsic speed of the network (vint) and the speed of the external drive (vext). Interestingly, we\n\ufb01nd that when vint > vext, the network bump leads the instant position of the moving stimulus for\nsuf\ufb01ciently weak stimuli, achieving anticipative tracking.\nWithout\n\u03b1 exp\nat time t and the speed of the moving stimulus is vext = dz0(t)/dt.\nDe\ufb01ne s = z(t) \u2212 z0(t) to be the displacement of the network bump relative to the external drive.\nWe consider that the network is able to track the moving stimulus, i.e., the network dynamics will\nreach a stationary state with dz(t)/dt = dz0(t)/dt and s a constant. Since we consider that the\nstimulus moves from left to right, s > 0 means that the network tracking is leading the moving\ninput; whereas s < 0 means the network tracking is lagging behind.\nUsing the Gaussian ansatz for the network state as given by Eqs. (7-9) and applying the projection\nmethod, we solve the network dynamics and obtain (see Supplementary Information),\n\n=\n, where \u03b1 represents the input strength, z0(t) is the stimulus\n\nI ext(x, t)\n\nexternal\n\ninput\n\nthe\n\nset\n\nbe\n\nto\n\n(\n\n)\n\n\u2212a +\n\nd = 2a\n\ns exp\n\n\u2212 s2\n8a2\n\n=\n\n1\n\u03b1\n\nAu\n\n\u03c4\nvext\n\n\u221a\n(\n\n)\n\na2 + (vext\u03c4v)2\nvext\u03c4v\nmd2\n\u03c4 \u03c4v\n\n\u2212 v2\n\next\n\n,\n\n.\n\n(13)\n\n(14)\n\next = md2/(\u03c4 \u03c4v), which\next < md2/(\u03c4 \u03c4v), which gives s > 0, i.e., the bump is leading\n\nCombining Eqs. (10, 13, 14), it can be checked that when vext = vint, v2\ngives s = 0 ; and when vext < vint, v2\nthe external drive (For detail, see Supplementary Information).\nFig. 3A presents the simulation result. There is a minor discrepancy between the theoretical pre-\ndiction and the simulation result:\nthe separation s = 0 happens at the point when the stimulus\nspeed vext is slightly smaller than the intrinsic speed of the network vint. This discrepancy arises\nfrom the distortion of the bump shape from Gaussian when the input strength is strong, the stimulus\nspeed is high and m is large, and hence the Gaussian ansatz on the network state is not accurate.\nNevertheless, for suf\ufb01ciently weak stimuli, the theoretical prediction is correct.\n\n4.1 Perfect tracking and perfect anticipative tracking\n\nAs observed in experiments, neural systems can compensate for time delays in two different ways:\n1) perfect tracking, in which the network bump has zero-lag with respect to the external drive, i.e.,\ns = 0; and 2) perfect anticipative tracking, in which the network bump leads the external drive by\napproximately a constant time tant = s/vext. In both cases, the tracking performance of the neural\nsystem is largely independent of the stimulus speed. We check whether a CANN with SFA exhibits\nthese appealing properties.\nDe\ufb01ne a scaled speed variable vext \u2261 \u03c4vvext/a. In a normal situation, vext \u226a 1. For instance,\ntaking the biologically plausible parameters \u03c4v = 100 ms and a = 50o, vext = 0.1 corresponds to\nvext = 500o/s, which is a rather high speed for a rodent rotating its head in ordinary life.\nBy using the scaled speed variable, Eq. (14) becomes\n(\u22121 +\n\n\u221a\n\n1 + v2\n\n(\n\n)\n\n[\n\n]\n\ns exp\n\n=\n\nAua\n\n4m\n\n\u2212 s2\n8a2\n\n1\n\u03b1\n\next)2\n\n\u2212 \u03c4\n\u03c4v\n\nvext\n\n.\n\n(15)\n\nv3\next\n\n6\n\n\fFigure 3: A. The separation s vs. the speed of the external input vext. Anticipative tracking s > 0\noccurs when vext < vint. The simulation was done with a network of N = 1000 neurons. The\nparameters are: J0 = 1, k = 0.1, a = 0.5, \u03c4 = 1, \u03c4v = 60, \u03b1 = 0.5 and m = 2.5\u03c4 /\u03c4v. B. An\nexample of anticipative tracking in the reference frame of the external drive. C. An example of\ndelayed tracking. In both cases, the pro\ufb01le of V (x, t) is lagging behind the bump U (x, t) due to\nslow SFA.\nIn the limit of vext \u226a 1 and consider s/(2\nAu\u03c4vvext(m \u2212 (cid:28)\n\n\u221a\n2a) \u226a 1 (which is true in practice), we get s \u2248\n\n)/\u03b1. Thus, we have the following two observations:\n\n(cid:28)v\n\n\u2022 Perfect tracking. When m \u2248 \u03c4 /\u03c4v, s \u2248 0 holds, and perfect tracking is effectively\nachieved. Notably, when there is no stimulus, m = \u03c4 /\u03c4v is the condition for the network\nstarting to have a traveling wave state.\n\u2022 Perfect anticipative tracking. When m > \u03c4 /\u03c4v, s increases linearly with vext, and the\nanticipative time tant is approximately a constant.\n\nThese two properties hold for a wide range of stimulus speed, as long as the approximation vext \u226a 1\nis applicable. We carried out simulations to con\ufb01rm the theoretical analysis, and the results are\npresented in Fig. 4. We see that: (1) when SFA is weak, i.e., m < \u03c4 /\u03c4v, the network tracking is\nlagging behind the external drive, i.e. s < 0 (Fig. 4A); (2) when the amplitude of SFA increases to a\ncritical value m = \u03c4 /\u03c4v, s becomes effectively zero for a wide range of stimulus speed, and perfect\ntracking is achieved (Fig. 4B); (3) when SFA is large enough satisfying m > \u03c4 /\u03c4v, s increases\nlinearly with vext for a wide range of stimulus speeds, achieving perfect anticipative tracking (Fig.\n4C); and (4) with the increasing amplitude of SFA, the anticipative time of the network also increases\n(Fig. 4D). Notably, by choosing the parameters properly, our model can replicate the experimental\n\ufb01nding on a constant leading time of around 25 ms when a rodent was tracking head-direction by\nADN cells (the red points in Fig. 4D for \u03c4 = 5 ms) [19].\n\n5 Conclusions\n\nIn the present study, we have proposed a simple yet effective mechanism to implement anticipative\ntracking in neural systems. The proposed strategy utilizes the property of SFA, a general feature\nin neuronal responses, whose contribution is to destabilize spatially localized attractor states in a\nnetwork. Analogous to asymmetrical neuronal interactions, SFA induces self-sustained travelling\nwave in a CANN. Compared to the former, SFA has the advantage of not requiring the hard-wired\nasymmetrical synapses between neurons. We systematically explored how the intrinsic mobility of\na CANN induced by SFA affects the network tracking performances, and found that: (1) when the\nintrinsic speed of the network (i.e., the speed of the travelling wave the network can support) is larger\nthan that of the external drive, anticipative tracking occurs; (2) an increase in the SFA amplitude can\nenhance the capability of a CANN to achieve an anticipative tracking with a longer anticipative time\nand (3) with the proper SFA amplitudes, the network can achieve either perfect tracking or perfect\nanticipative tracking for a wide range of stimulus speed.\nThe key point for SFA achieving anticipative tracking in a CANN is that it provides a negative feed-\nback modulation to destabilize strong localized neuronal responses. Thus, other negative feedback\n\n7\n\n(cid:34)(cid:35)(cid:36)00.0040.0080.0120.016\u22120.06\u22120.0300.030.06vextS vint(cid:14)\u03c0\u03c0/20\u03c0/2\u03c0s>0dU(x)Iext(x)V(x)(cid:14)(cid:14)\u03c0\u03c0/20\u03c0/2\u03c0s<0dIext(x)U(x)V(x)(cid:14)(cid:14)\fFigure 4: Tracking performances of a CANN with SFA. A. An example of delayed tracking for\nm < \u03c4 /\u03c4v; B. An example of perfect tracking for m = \u03c4 /\u03c4v. s = 0 roughly holds for a wide range\nof stimulus speed. C. An example of perfect anticipative tracking for m > \u03c4 /\u03c4v. s increases linearly\nwith vext for a wide range of stimulus speed. D. Anticipative time increases with the SFA amplitude\nm. The other parameters are the same as those in Fig. 3.\n\nmodulation processes, such as short-term synaptic depression (STD) [20, 21] and negative feedback\nconnections (NFC) from other networks [22], should also be able to realize anticipative tracking.\nIndeed, it was found in the previous studies that a CANN with STD or NFC can produce leading\nbehaviors in response to moving inputs. The three mechanisms, however, have different time scales\nand operation levels: SFA has a time scale of one hundred milliseconds and functions at the single\nneuron level; STD has a time scale of hundreds to thousands of milliseconds and functions at the\nsynapse level; and NFC has a time scale of tens of milliseconds and functions at the network level.\nThe brain may employ them for different computational tasks in conjunction with brain functions.\nIt was known previously that a CANN with SFA can retain travelling wave [12]. But, to our knowl-\nedge, our study is the \ufb01rst one that links this intrinsic mobility of the network to the tracking per-\nformance of the neural system. We demonstrate that through regulating the SFA amplitude, a neural\nsystem can implement anticipative tracking with a range of anticipative times. Thus, it provides\na \ufb02exible mechanism to compensate for a range of delay times, serving different computational\npurposes, e.g., by adjusting the SFA amplitudes, neural circuits along the hierarchy of a signal trans-\nmission pathway can produce increasing anticipative times, which compensate for the accumulated\ntime delays. Our study sheds light on our understanding of how the brain processes motion infor-\nmation in a timely manner.\n\nAcknowledgments\n\nThis work is supported by grants from National Key Basic Research Program of China\n(NO.2014CB846101, S.W.), and National Foundation of Natural Science of China (No.11305112,\nY.Y.M.; No. 31261160495, S.W.), and the Fundamental Research Funds for the central Universities\n(No. 31221003, S. W.), and SRFDP (No.20130003110022, S.W), and Research Grants Council\nof Hong Kong (Nos. 605813, 604512 and N HKUST606/12, C.C.A.F. and K.Y.W), and Natural\nScience Foundation of Jiangsu Province BK20130282.\n\n8\n\n(cid:34)(cid:35)(cid:36)(cid:37)00.10.20.30.4\u22120.02\u22120.015\u22120.010\u22120.02vextS m = 0.5 \u03c4/\u03c4v00.10.20.30.4\u22120.02\u22120.0100.01vextS S = 0m = \u03c4/\u03c4v00.10.20.30.400.010.020.030.04S = vext tantvextS m = 2.5 \u03c4/\u03c4v00.10.20.30.405101520m = 1.5 \u03c4/\u03c4vm = 2.0 \u03c4/\u03c4vm = 2.5 \u03c4/\u03c4vvexttant [\u03c4] \fReferences\n[1] L. G. Nowak, M. H. J. Munk, P. Girard & J. Bullier. Visual Latencies in Areas V1 and V2 of\n\nthe Macaque Monkey. Vis. Neurosci., 12, 371 \u2013 384 (1995).\n\n[2] C. Koch, M. Rapp & Idan Segev. A Brief History of Time (Constants). Cereb. Cortex, 6, 93 \u2013\n\n101 (1996).\n\n[3] H. T. Blair & P. E. Sharp. Anticipatory Head Direction Signals in Anterior Thalamus: Evidence\nfor a Thalamocortical Circuit that Integrates Angular Head Motion to Compute Head Direction.\nJ. Neurosci., 15(9), 6260 \u2013 6270 (1995).\n\n[4] J. S. Taube & R. U. Muller. Comparisons of Head Direction Cell Activity in the Postsubiculum\n\nand Anterior Thalamus of Freely Moving Rats. Hippocampus, 8, 87 \u2013 108 (1998).\n\n[5] J. P. Bassett, M. B. Zugaro, G. M. Muir, E. J. Golob, R. U. Muller & J. S. Taube. Passive\nMovements of the Head Do Not Abolish Anticipatory Firing Properties of Head Direction\nCells. J. Neurophysiol., 93, 1304 \u2013 1316 (2005).\n\n[6] R. Nijhawan. Motion Extrapolation in Catching. Nature, 370, 256 \u2013 257 (1994).\n[7] J. R. Duhamel, C. L. Colby & M. E. Goldberg. The Updating of the Representation of Visual\n\nSpace in Parietal Cortex by Intended Eye Movements. Science 255, 90 \u2013 92 (1992).\n\n[8] R. Nijhawan & S. Wu. Phil. Compensating Time Delays with Neural Predictions: Are Predic-\n\ntions Sensory or Motor? Trans. R. Soc. A, 367, 1063 \u2013 1078 (2009).\n\n[9] P. E. Sharp, A. Tinkelman & Cho J. Angular Velocity and Head Direction Signals Recorded\nfrom the Dorsal Tegmental Nucleus of Gudden in the Rat: Implications for Path Integration in\nthe Head Direction Cell Circuit. Behav. Neurosci., 115, 571 \u2013 588 (2001).\n\n[10] B. Gutkin & F. Zeldenrust. Spike Frequency Adaptation. Scholarpedia, 9, 30643 (2014).\n[11] J. Benda & A. V. M. Herz. A Universal Model for Spike-Frequency Adaptation. Neural Com-\n\nput., 15, 2523 \u2013 2564 (2003).\n\n[12] P. C. Bressloff. Spatiotemporal Dynamics of Continuum Neural Fields. J. Phys. A, 45, 033001\n\n(2012).\n\n[13] R. Ben-Yishai, R. L. Bar-Or & H. Sompolinsky. Theory of Orientation Tuning in Visual Cortex.\n\nProc. Natl. Acad. Sci. U.S.A., 92, 3844 \u2013 3848 (1995).\n\n[14] K. Zhang. Representation of Spatial Orientation by the Intrinsic Dynamics of the Head-\n\nDirection Cell Ensemble: a Theory. J. Neurosci., 16, 2112 \u2013 2126 (1996).\n\n[15] A. P. Georgopoulos, M. Taira & A. Lukashin. Cognitive Neurophysiology of the Motor Cortex.\n\nScience, 260, 47 \u2013 52 (1993).\n\n[16] A. Samsonovich & B. L. McNaughton. Path Integration and Cognitive Mapping in a Continu-\n\nous Attractor Neural Network Model. J. Neurosci, 17, 5900 \u2013 5920 (1997).\n\n[17] K. Wimmer, D. Q. Nykamp, C. Constantinidis & A. Compte. Bump Attractor Dynamics in\nPrefrontal Cortex Explains Behavioral Precision in Spatial Working Memory. Nature, 17(3),\n431 \u2013 439 (2014).\n\n[18] C. C. A. Fung, K. Y. M. Wong & S. Wu. A Moving Bump in a Continuous Manifold: a\nComprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks.\nNeural Comput., 22, 752 \u2013 792 (2010).\n\n[19] J. P. Goodridge & D. S. Touretzky. Modeling attractor deformation in the rodent head direction\n\nsystem. J. Neurophysio., 83, 3402 \u2013 3410 (2000).\n\n[20] C. C. A. Fung, K. Y. M. Wong, H. Wang & S. Wu. Dynamical Synapses Enhance Neural\nInformation Processing: Gracefulness, Accuracy, and Mobility. Neural Comput., 24, 1147 \u2013\n1185 (2012).\n\n[21] C. C. A. Fung, K. Y. M. Wong & S. Wu. Delay Compensation with Dynamical Synapses. Adv.\nin NIPS. 25, P. Bartlett, F. C. N. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger\n(eds), 1097 \u2013 1105 (2012).\n\n[22] W. Zhang & S. Wu. Neural Information Processing with Feedback Modulations. Neural Com-\n\nput., 24(7), 1695 \u2013 1721 (2012).\n\n9\n\n\f", "award": [], "sourceid": 334, "authors": [{"given_name": "Yuanyuan", "family_name": "Mi", "institution": "Weizmann Institute of Science"}, {"given_name": "C. C. Alan", "family_name": "Fung", "institution": "The Hong Kong University of Science and Technology"}, {"given_name": "K. Y. Michael", "family_name": "Wong", "institution": "Department of Physics, Hong Kong University of Science and Technology"}, {"given_name": "Si", "family_name": "Wu", "institution": "Beijing Normal University"}]}