{"title": "Temporal Coding using the Response Properties of Spiking Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 1457, "page_last": 1464, "abstract": null, "full_text": "Temporal Coding using the Response Properties of Spiking Neurons\n\nThomas Voegtlin INRIA - Campus Scientifique, B.P. 239 F-54506 Vandoeuvre-Les-Nancy Cedex, FRANCE voegtlin@loria.fr\n\nAbstract\nIn biological neurons, the timing of a spike depends on the timing of synaptic currents, in a way that is classically described by the Phase Response Curve. This has implications for temporal coding: an action potential that arrives on a synapse has an implicit meaning, that depends on the position of the postsynaptic neuron on the firing cycle. Here we show that this implicit code can be used to perform computations. Using theta neurons, we derive a spike-timing dependent learning rule from an error criterion. We demonstrate how to train an a uto-encoder neural network using this rule.\n\n1\n\nIntroduction\n\nThe temporal coding hypothesis states that information is encoded in the precise timing of action potentials sent by neurons. In order to achieve computations in the time domain, it is thus necessary to have neurons spike at desired times. However, at a more fun damental level, it is also necessary to describe how the timings of action potentials received by a neuron are combined together, in a way that is consistent with the neural code. So far, the main theory has posited that the shape of post-synaptic potentials (PSPs) is relevant for computations [1, 2, 3]. In these models, the membrane potential at the soma of a neuron is a weighted sum of PSPs arriving from dendrites at different ti mes. The spike time of the neuron is defined as the time when its membrane potential first reaches a firing threshold, and it depends on the precise temporal arrangement of PSPs, thus enabling com putations in the time domain. Hence, the nature of the temporal code is closely tied to the shape of PSPs. A consequence is that the length of the rising segment of post-synaptic potentials limits the available coding interval [1, 2]. Here we propose a new theory, based on the non-linear dynamic s of integrate-and-fire neurons. This theory takes advantage of the fact that the effect of synaptic currents depends on the internal state of the postsynaptic neuron. For neurons spiking regularly, th is dependency is classically described by the Phase Response Curve (PRC) [4]. We use theta neurons, which are mathematically equivalent to quadratic integrate-and-fire neurons [5, 6]. In these neu ron models, once the potential has crossed the firing threshold, the neuron is still sensitive to incomi ng currents, which may change the timing of the next spike. In the proposed model, computations do not rely on the shape of PSPs, which alleviates the restriction imposed by the length of their rising segment. The refore, we may use a simplified model of synaptic currents; we model synaptic currents as Diracs, which means that we do not take into account synaptic time constants. Another advantage of our model is that computations do not rely on the delays imposed by inter-neuron transmission; this me ans that it is not necessary to fine-tune delays in order to learn desired spike times.\n\n\f\n2\n2.1\n\nDescription of the model\nThe Theta Neuron\n\nThe theta neuron is described by the following differential equation: d = (1 - cos) + I (1 + cos) dt (1)\n\nwhere is the \"potential\" of the neuron, and I is a variable input current, measured in radians per unit of time. For convenience, we call units of time 'milliseconds'. The neuron is said to fire everytime crosses . The dynamics of the model can be represented on a phase circl e (Figure 1). The effect of an input current is not uniform across the circl e; currents that occur late (for close to ) have little effect on , while currents that arrive when is close to zero have a much greater effect.\n+ 0\n\n\n- 0\n\n I>0\n\nI<0\n\nFigure 1: Phase circle of the theta model. The neuron fires everytime crosses . For I < 0 there + + - 1+ are two fixed points: An unstable point 0 = arccos 1-I , and an attractor 0 = -0 . I 2.2 Synaptic interactions\n\nThe input current I is the sum of a constant current I0 and transient synaptic currents Ii (t), where i 1..N indexes the synapses: iN Ii (t) (2) I = I0 +\n=1\n\nSynaptic currents are modeled as Diracs : Ii (t) = wi (t - ti ), where ti is the firing time of presynaptic neuron i, and wi is the weight of the synapse. Transmission delays are not taken into account.\n\nFigure 2: Response properties of the theta model. Curves shows the change of firing time tf of a neuron receiving a Dirac current of weight w at time t. Left: For I0 > 0, the neuron spikes regularly (I0 = 0.005, (0) = - ). If w is small, the curves corresponding to w > 0 and w < 0 are symmetric; the positive curve is called the Phase Response Curve (PRC). If w is large, curves are no longer symmetric; the portions correspond to the ascending (resp. descending) phase of sin have different slopes. Right: Response for I0 < 0. The initial condition is slightly above the unstable + equilibrium point (I0 = -0.005, (0) = 0 + 0.0001), so that the neuron fires if not perturbed. For w > 0, the response curve is approximately linear, until it reach es zero. For w < 0, the current might cancel the spike if it occurs early.\n\n\f\nFigure 2 shows how the firing time of a theta neuron changes wit h the time of arrival of a synaptic current. In our time coding model, we view this curve as the tr ansfer function of the neuron; it describes how the neuron converts input spike times into out put spike times. 2.3 Learning rule\n\nWe derive a spike-timing dependent learning rule from the ob jective of learning a set of target firing times. Following [2], we consider the mean squared error, E , between desired spike times ts and actual spike times ts : E =< (ts - ts )2 > (3) where < . > denotes the mean. Gradient descent on E yields the following stochastic learning rule: wi = - The partial derivative\n ts wi\n\nE ts = -2 (ts - ts ) wi wi\n\n(4)\n\nexpresses the credit assignment problem for synapses.\n d + i i-\n\n+ i\n\nwi ti\n\n\n\nts\n\nti\n\ntime\n\nFigure 3: Notations used in the text. An incoming spike triggers an instantaneous change of the - + potential . i (resp. i ) denotes the postsynaptic potential before (resp. after) t he presynaptic + spike. A small modification dwi of the synaptic weight wi induces a change di Let F denote the \"remaining time\", that is, the time that remains before the neuron will fire: d F (t) = (1 - cos) + I (1 + cos) (t)\n\n(5)\n\nIn our model, I is not continuous, because of Dirac synaptic currents. For t he moment, we assume + that is between the unstable point 0 and . In addition, we assume that the neuron receives one spike on each of its synapses, and that all synaptic weights are positive. Let tj denote the time of - + arrival of the action potential on synapse j . Let j (resp. j ) denote the potential before (resp. after) the synaptic current: - - j = (tj ) (6) - - + + j = (tj ) = j + wj (1 + cos j ) We consider the effect of a small change of weight wi . We shall rewrite integral (5) on the intervals where the integrand is continuous. To keep notations simple, we assume that action potentials are - ordered, ie : tj tj +1 for all j . For consistency, we use the notation N +1 = . We may write: F (ti ) = j\n- j +1\n\ni\n\n+ j\n\nd (1 - cos) + I0 (1 + cos)\n\n(7)\n\nThe partial derivative of the spiking time ts can be expressed as : + - + j F i F j F j ts =+ + +- + wi i wi j wi j wi >i\n\n(\n\n8)\n\n\f\nIn this expression, the sum expresses how a change of weight wi will modify the effect of other spikes, for j > i. The j th terms of this sum depend on the time elapsed between tj and ti . Since we have no a priori information on the distribution of tj given ti , we shall consider that this term is not E correlated with wi . For that reason, we neglect this sum in our stochastic learn ing rule: F + ts +i wi i wi which yields :\n- (1 + cos i ) ts - + + wi (1 - cos i ) + I0 (1 + cos i )\n\n(9)\n\n(10)\n\n+ + Note that this expression is not bounded when i is close to the unstable point 0 . In that case, is in a region where it changes very slowly, and the timing of o ther action potentials for j > i will mostly determine the firing time ts . This means that approximation (9) will not hold. In addition, + -+ it is necessary to extend the learning rule to the case i [0 0 [, where the above expression is negative. For these reasons, we introduce a credit bound, C , and we modify the learning rule as follows: ts ts if 0 < - < C then: wi = -2 (ts - ts ) (11) wi wi else: wi = 2 (ts - ts )C (12)\n\n2.4\n\nAlgorithm\n\nThe algorithm updates the weights in the direction of the gra dient. The learning rule takes effect at the end of a trial of fixed duration. If a neuron does not fire at all during the tri al, then its firing time is considered to be equal to the duration of the trial. For each synapse, it is necessary to compute the credit from E quation (10) everytime a current is transmitted. We may relax the assumption that each synapse receives one single action potential; if a presynaptic neuron fires several times before the postsyna ptic neuron fires, then the credit corresponding to all spikes is summed. Theta neurons were simulated using Euler integration of Equ ation (1). The time step must be carefully chosen; if the temporal resolution is too coarse, then the credit assignment problem becomes too difficult, which increases the number of trials necessar y for learning. On the other hand, small values of the time step mean that simulations take more time.\n\n3\n\nAuto-encoder network\n\nPredicting neural activities has been proposed as a possible role for spike-timing dependent learning rules [7]. Here we train a network to predict its own activiti es using the learning rule derived above. For this, a time-delayed version of the input (echo) is used as the desired output (see Figure 4). The network has to find a representation of the input that minimiz es mean squared reconstruction error. The network has three populations of neurons: (i) An input po pulation X of size n neurons, where an input vector is represented using spike times. We call Inter Stimulus Interval (ISI) the interval between the spikes encoding the input and the echo. After the ISI, population X fires a second burst of spikes, that is a time-delayed version of the initial burst. (ii) An output population Y , of size m neurons, that is activated by neurons in X . (iii) A population X of size n neurons, where the input is reconstructed. Neurons in X are activated by Y . The learning rule updates the feedback connections (wij )in,j m from Y to X , comparing spike times in X and in X . We use I0 < 0, so the response to positive transient currents is approxim ately linear (see fig. 2). We thus expect neurons to perform linear summation of spike tim es. For the feed-forward connections from X to Y , we use the transpose of the feedback weights matrix. This is inspired by Oja's Principal Subspace Network [8]. If spike times are within the linear part of the response curve, then we expect this network to perform Principal Component Analy sis (PCA) in the time domain. However, one difference is that the PRC we use is always positive (type I neurons). This means that spike times can only code for positive values (even though synaptic weights can be of both signs).\n\n\f\nOutput Y feedback feed-forward X\n\nEcho Input time\n\nFigure 4: Auto-encoder network. An input vector is translated into firing times of the input po pulation. Output neurons are activated by input neurons thro ugh feed-forward connections. A reconstruction of the input burst is generated through feedba ck connections. Target firing times are provided by a delayed version of the input burst (echo). In order to code for values of both signs, one would need a tran sfer function that changes its sign around a time that would code for zero, so that the effect of a c urrent is reversed when its arrival time crosses zero. Here we may view the neural code as a positive code: Early spikes code for high values, and late spikes code for values close to zero. In this architecture, it is necessary to ensure that each neu ron in Y fires a single spike on each trial. In order to do this, we impose that neurons in Y have the same average firing time. For this, we add a centering term to the learning rule: wij = - E - j wij (13)\n\nThis modification of the learning rule results in neurons tha t have no preferred firing order.\n\nwhere I and j is the average phase of neuron j . j is a leaky average of the difference R between the firing time tj and the average firing times of all neurons in population Y . It is updated after each trial: t ( m 1k j j + (1 - ) j - 14) tk m\n=1\n\n4\n\nExperiments\n\nWe used I0 = -0.01 for all neurons. This ensures that neurons have no spontaneo us activity. At the beginning of a trial, all neurons were initialized to the ir stable fixed point. In order to balance the effect of the different sizes of populations X and Y , different values of were used for X and Y neurons: We used X = 0.1 and Y = m X . In the leaky average we used = 0.1 n In each experiment, the input vector was encoded in spike times. When doing so, one must make sure that the values taken by the input are within the coding interval of the neurons, ie the range of values where the PRC is not zero. In practice, spikes that arrive too late in the firing cycle are not taken into account by the learning rule. In that case, the weights corresponding to other synapses become overly increased, which eventually causes some postsynaptic neurons in X to fire before presynaptic neurons in Y (\"anticausal spikes\"). If this occurs, one possibility is to reduce the variance of the input. 4.1 Principal Component Analysis of a Gaussian distribution\n\nA two-dimensional Gaussian random variable was encoded in the spike times of three input neurons. The ellipsoid had a long axis of standard deviation 1ms and a short axis of deviation 0.5ms, and it\n\n\f\nwas rotated by /3. Because the network does not have an absolute time reference, it is necessary to use three input neurons, in order to encode two degrees of f reedom in relative spiking times. The output layer had two neurons (one degree of freedom). The refore the network has to find a 1D representation of a 2D variable, that minimizes the mean-squared reconstruction error. The input was encoded as follows: t 0=3 t1 = 3 + 1 cos( /3) + 0.52 sin( /3) (15) t2 = 3 + 0.52 cos( /3) + 1 sin( /3) where 1 and 2 are two independent random variables picked from a Gaussian distribution of variance 1. Input spikes times were centered around t = 3ms, where t = 0 denotes the beginning of a trial. We used a time step of 0.05 ms. Each trial lasted for 400 iterations, which corresponds to 20ms of simulated time. The ISI was 5ms. The credit bound was C = 1000. Other parameters were = 0.0001 and = 0.001. Weights were initialized with random values between 0.5 and 1.5.\n\nFigure 5: Principal Component Analysis of a 2D Gaussian distribution. The input vector was encoded in the relative spike times of three input neurons. Top: Evolution of the weights over 20.000 learning iterations. Bottom: Final synaptic weights repre sented as bars. Note the complementary shapes of weight vectors. Right: The input (white dots) and its reconstruction (dark dots) from the network's activities. Each branch corresponds to a firing or der of the two output neurons. Figure 5 shows that the network has learned to extract the pri ncipal direction of the distribution. Two branches are visible in the distribution of dots correspond ing to the reconstruction. They correspond to two firing orders of the output neurons. The direction of th e branches results from the synaptic weights of the neurons. Note that the lower branch has a sligh t curvature. This suggests that the response function of neurons is not perfectly linear in the i nterval where spike times are coded. The fact that branches do not exactly have the same orientation might result from non-linearities, or from the approximation made in deriving the learning rule. There are six synaptic weights in the network. One degree of f reedom per neuron in X is used to adapt its mean firing times to the value imposed by the ISI; the smaller the ISI, the larger the weights. This \"normalization\" removes three degrees of freedom. One additional constraint is imposed by the centering term that was added to the learning rule in (13) . Thus the network had two degrees of freedom. It used them to find the directions of the two branches shown in Figure 5 (left). These two branches can be viewed as the base vectors used in the compres sed representation in Y . The network uses two base vectors in order to represent one si ngle principal direction; each codes for one half of the Gaussian. This is because the network uses a positive code, where negative values are not allowed. 4.2 Encoding natural images\n\nAn encoder network was trained on the set of raw natural image s used in [9]1 . The encoder had 64 output neurons and 256 input neurons. On each trial, a random patch of size 16 16 was extracted\n1\n\nImages were retrieved from http://redwood.berkeley.edu/bruno/spar senet/\n\n\f\nfrom a random image of the dataset, and encoded in the network . Raw grey values from the dataset were encoded as milliseconds. The standard deviation per pi xel was 1.00ms. The time step of the simulation was 0.1ms, and each trial lasted for 200 time steps (20ms). The ISI was 9ms, and the parameters of the learning rule were = 0.0001, C = 50 and = 0.001. Weights were initialized with random values between 0 and 0.3.\n\nFigure 6: Synaptic weights learned by the network. 64 neurons were trained to represent natural images patches of size 16 16. Different grey scales are used in order to display positive and negative weights (black is negative, white is positive). Left: grey scale between -1 and 1. Only positive weights are visible at this scale, because they are much larger than negative weights. Right: grey scale between -0.1 and 0.1. Negative weights are visible, positive weights are beyond scale. Synaptic weights after 100.000 trials are shown in Figure 6. There is a strong difference of a mplitude between positive and negative weights; positive weights typically have values between 0 and 1, while negative weights are one order of magnitude smaller. For tha t reason, weights are displayed twice, with two different grey scales. An image reconstructed from spike times is shown in Figure 7. After training, the mean reconstruction error on the entire datas et was 0.25ms/pixel. For comparison, the mean error performed by Oja's principal subspace network [8 ] trained on the same image patches was 0.11ms/pixel. The difference of amplitude between positive and negative weights results from higher sensitivity of the response curves to negative weights, as shown in Figure 2. Synaptic weights with negative values have the ability to strongly delay the output spike, and even to cancel it. Synaptic weights have the shape of local filters, with antagonistic center-surround structures. This contrasts with the base vectors typically obtained from PCA of natural images, which are not local. One possible explanation lies in the response properties of the theta neurons. The response function is not linear, especially in the case of negative weights (Figure 2). This will disfavor solutions involving linear combinations of both positive and negative weights, and favor sparse representations. Hence, the network could be performing something similar to Nonlinear PCA [10].\n\n5\n\nConclusions\n\nWe have shown that the dynamic response properties of spikin g neurons can be effectively used as transfer functions, in order to perform computations (in th is paper, PCA and Nonlinear PCA). A similar proposal was made in [11], where the PRC of neurons ha s been adapted to a biologically realistic STDP rule. Here we took a complementary approach, adapting the learning rule to the neuronal dynamics. We used theta neurons, which are of type I, and equivalent to quadratic integrate-and-fire neurons. Type I neurons have a PRC that is always positive. This means that spike times can encode only\n\n\f\nFigure 7: Natural image and reconstruction from spike times. The 512 512 image from the training set (left) was divided into 16 16 patches, and encoded using 64 neurons. The reconstruction (right) is derived from spikes times in X . Standard deviation of the encoded images was 1.00ms/pixel. The mean reconstruction error on the entire dataset was 0.25ms/pixel, about 2.5 times the error made by PCA. positive values. In order to encode values of both signs, one would need the transfer function to change its sign around a time that codes for zero. This will be possible with more complex type II neurons, where the sign of the PRC is not constant. Acknowledgments The author thanks Samuel McKennoch and Dominique Martinez for helpful comments.\n\nReferences\n[1] W. Maass. Lower bounds for the computational power of networks of spiking neurons. Neural Computation, 8(1):140, 1996. [2] S.M. Bohte, J.N. Kok, and H. La Poutre. Spike-prop: error-backprogation in multi-layer networks of spiking neurons. Neurocomputing, 48:1737, 2002. [3] A. J. Bell and L. C. 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Sparse coding of natural images produces localized, oriented, bandpass receptive fields. Nature, 381:607609, 1996. [10] E. Oja. The nonlinear PCA learning rule in independent component analysis. Neurocomputing, 17(1):25 46, 1997. [11] Lengyel M., Kwag J., Paulsen O., and Dayan P. Matching storage and recall:hippocampal spike timingdependent plasticity and phase response curves. Nature Neuroscience, 8:16771683, 2006.\n\n\f\n", "award": [], "sourceid": 3123, "authors": [{"given_name": "Thomas", "family_name": "Voegtlin", "institution": null}]}