Coherence detection in a spiking neuron via Hebbian learning

Transcription

1 Neurocomputing (2002) Coherence detection in a spiking neuron via Hebbian learning L. Perrinet, M. Samuelides ONERA-DTIM, 2 Av. E. Belin, BP 4025, Toulouse, France Abstract It is generally assumehat neurons communicate through temporal ring patterns. As a rst step, we will study the learning of a layer of realistic neurons in the particular case where the relevant messages are formedby temporally correlatedpatterns, or synre patterns. The model is a layer of integrate-and-re neurons with synaptic current dynamics that adapts by minimizing a cost according to a gradient descent scheme. The cost we dene leads to a rule similar to spike-time dependent Hebbian plasticity. Moreover, our results show that the rule that we derive is biologically plausible and leads to the detection of the coherence in the input in an unsupervisedway. An application to shape recognition is shown as an illustration. c 2002 Publishedby Elsevier Science B.V. Keywords: Spiking neural networks; Hebb rule; Spike time dependent Hebbian plasticity; Gradient descent 1. Description of the model 1.1. Coding scheme We will represent (as in [2]) the signal S i at synapse i by the sum of Dirac pulses locatedat the spiking times ti k drawn from the lists of spikes i (see Fig. 1-left). S i = (t ti k ): (1) k i Synre patterns are generatedin analogy with the response of a retina to ashedbinary images. The input of the synapses is characterizedas the output of single-synapse IF Corresponding author. address: perrinet@cert.fr (L. Perrinet). URL: /02/$ - see front matter c 2002 Publishedby Elsevier Science B.V. PII: S (02)

2 134 L. Perrinet, M. Samuelides / Neurocomputing (2002) Fig. 1. Neuron model: (left) Input spikes (with a synre pattern at t = 25 ms), are (middle) modulated in time andamplitude forming postsynaptic current pulses andare nally (right) integratedat the soma. When the potential (plain line) reaches the threshold(dottedline), a spike is emittedanhe potential is decreased. A sample PSP (synapse 1) is also shown. neurons responding to a specic binary input. This response may be described as the sum of two random point processes with dierent time scales. At a narrow time scale, the input is the spontaneous activity, i.e. a background noise independent of time and synapses that may be described by a Poisson point process of rate 1= noise. At a larger time scale, the synre pattern activates a given subset M of synapses once per ash with a temporal correlation dened by its jitter jitter (see Fig. 1-left) Integrate-and-re layer We will consider N 1 synapses (indexed by i) connecteo a layer of N 2 neurons j. Those are generalizedversion of IF neurons with synaptic current dynamics, a one compartment model with no delay and the synapses have contacts characterized by their weight w ji. The state variables are N 1 N 2 synaptic driving conductances g ji and N 2 membrane potentials V j. Incoming spikes trigger those conductances by opening the driving gates with time constant g : dg ji = 1 g g ji + w ji S i (2) anhe potential V j at the soma integrates with time constant V the driving currents anhe leaking current g leak (with a potential V rest 70 mv): (see Fig. 1-

3 middle): V dv j L. Perrinet, M. Samuelides / Neurocomputing (2002) = g leak (V rest V j )+ ( 16i6N 2 g ji ) : (3) When V j reaches the threshold s potential V threshold 54 mv from below, the target neuron res (see Fig. 1-right) andis shunted(v j is set e.g. to V reset 75 mv) Reduced equations We introduce reduced equations for this IF Layer to study its dynamical behavior andsimplify its implementation. In fact, this reduction follows the concept of the spike-response model (SRM) which was extensively studied in [2,3]. It is similar to [4] which aimed at reducing the STDHP equations to a set of rst-order equations. dc i = 1 c i + S i ; g dp i V = p i + c i (4) then, V j =V rest +( 16i6N 1 w ji p i ) veries the equation system (2), (3), with g ji =w ji c i. To account for the thresholdmechanism at a spiking time tj k, we may then add a resetting value to V j by setting j (tj k)=v reset V threshold anhen j (t) V = j (t): (5) So that nally, an equivalent version of the IF Layer consists of (4), (5) and ( ) V j (t)=v rest + w ji p i (t) + j (t): (6) 16i6N 1 This formulation depends only on the present state and not on the past values. It is therefore biologically more plausible andcomputationally cheaper. Integrating these equations after emission of a presynaptic spike at t i or a postsynaptic spike at t j leads to ( c i (t) = exp t t ) i ; (7) g p i (t)= ( ( g exp t t i V g g ( (t)=(v reset V threshold ) exp ) ( exp t t i t t j V v )) ; (8) ) : (9) Those equations (6) (9) are the equivalent SRM version of our IF model. More precisely Eq. (7) represents the PostSynaptic Current (PSC), see Fig. 1-middle, and Eq. (8)

4 136 L. Perrinet, M. Samuelides / Neurocomputing (2002) the PostSynaptic Potential (PSP), see Fig. 1-right, which may be experimentally observed. 2. The learning mechanism 2.1. Denition of the cost function Basedon neurophysiological studies, we set the following principles: (1) the learning is associatedwith a spiking response: the nth learning step occurs at the nth output ring time t n, (2) to discriminate between the dierent input patterns, the output voltage should be close to a winner-take-all conguration: the potential of the winning neuron (which we index j = j n ) shouldbe above thresholdwhereas other neurons shouldbe hyperpolarized, (3) economy of the total synaptic ecacy andcurrent use shouldbe respected. A possible cost function may therefore be the squareddistance to the potentials of neurons at the ring time t n added to the total sum of the squared weights: 2E = (V j Vj t ) 2 + ( ) ji; 2 (10) 16j6N 2 16i6N 1 16j6N 2 V t j = V rest for j j n ; (11) Vj t n = V threshold +V; (12) where and are scaling parameters andwe set V 5mV Gradient descent It follows from Eqs. (10) =(V j Vj t j =(V j V t j )p i j + w + w ji : We may therefore formulate the gradient descent algorithm in our model as wji n+1 = wji ji =(1 n )wji n + n (Vj t V j j + ji V ji

5 L. Perrinet, M. Samuelides / Neurocomputing (2002) Fig. 2. STDHP: relative weight change versus time dierence between input and output spikes in a single neuron modelized with our model replicating the experimental conditions of [1]. with learning factors n ; n and n which satises (e.g. for n ) n=1::: n and n=1::: 2 n. Finally, writing ṗ i =dp i =, w n+1 ji =(1 n )w n ji + n (V t j V j )p i + n ( wji ṗ i ) ṗ i : (13) 2.3. Spike-time dependent plasticity A closer look at Eq. (13) shows that a direction in the change of w ji is proportional to (V target j V j )p i. This is the Hebbian part of the rule: when a neuron j res after the ring of synapse j, there is a mechanism that strengthen the connection. The strengthening depends therefore on the relative time of the pre- and post-synaptic spikes (see Fig. 2) as is observedin biological systems [1]. 3. Numerical results We implemented this model using discrete versions of the dierential equations (forwardeuler method) on a MATLAB system Response to synre patterns To achieve this experiment we presentedsynre patterns to the layer. The weights were set at random so that the network could re to all the inputs. The patterns were presented at random times that were suciently distant. This unsupervised learning converges quickly, andas may be observedin neuromuscular connectivity, the synapses teno sparsify anhe neurons teno respono only one input (see Fig. 3).

6 138 L. Perrinet, M. Samuelides / Neurocomputing (2002) Fig. 3. Coherence detection: (left) dierent input patterns (ring times t = 100; 300; 500; 700; 900 ms) are (right) learnt by the system: only one neuron per input res (100 learning steps). Fig. 4. Orientedbars detection: after learning, the weights show sensitivity to orientation (black:off; white:on; gray:neutral) Response to oriented bars The next experiment consistedin applying those results to a basic retina which input consists of centeredrotatedlines. A xedanalogical contrast layer (ON andoff radial cells) sends then spikes to the learning layer that adapts with the rule we presented. We observe unsupervisedemergence of V1-like receptors elds sensitive to the orientation (see Fig. 4). Further experiments with lateral interactions and accounting for dendritic delay show even more realistic lters and column architecture. 4. Conclusion We have presentedan original gradient descent methoo nda learning rule for a layer of spiking neurons. The simplicity of the rule gives a new insight into the comprehension of the mechanism behinhe observedstdhp. Further work is done for the detection of asynchronous patterns.

7 L. Perrinet, M. Samuelides / Neurocomputing (2002) However, this study should be extended to more realistic spike trains (e.g. bursts), account for more complex behavior (e.g. facilitation anddepression) andmay be extendeo population of neurons andrecurrent systems. Acknowledgements This work has been initiated during the EU Advanced Course in Computational Neuroscience. LP wish to thank its organizers, the teachers, the course-mates andmy tutor, S. Panzeri. References [1] G.-Q. Bi, M.-M. Poo, Synaptic modications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, andpostsynaptic cell type, J. Neurosci. 18 (1998) [2] W. Gerstner, Spiking neurons, in: W. Maass, C.M. Bishop (Eds.), PulsedCoupledNeural Networks, MIT Press, Cambridge, MA, 1999, pp (Chapter 1). [3] R. Kempter, W. Gerstner, J. van Hemmen, Hebbian learning andspiking neurons, Phys. Rev. E 59 (4) (1999) [4] L. Perrinet, A. Delorme, S. Thorpe, Network of integrate-and-re neurons using rank order coding A: how to implement spike timing dependent plasticity, Neurocomputing (2001) Laurent Perrinet is a Ph.D. student in Computational Neuroscience, under the direction of Manuel Samuelides at the CERT-ONERA, Toulouse and in narrow collaboration with the team of Simon Thorpe at the CERCO-CNRS, Toulouse. He works on theoretical andsimulatedaspects of neural coding, especially on the implication of fast-categorization visual experiments. Working areas span learning (especially in spike timing dependant plasticity) and the statistics of natural images.