nips nips2004 nips2004-28 nips2004-28-reference knowledge-graph by maker-knowledge-mining
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Author: Sophie Deneve
Abstract: We propose a new interpretation of spiking neurons as Bayesian integrators accumulating evidence over time about events in the external world or the body, and communicating to other neurons their certainties about these events. In this model, spikes signal the occurrence of new information, i.e. what cannot be predicted from the past activity. As a result, firing statistics are close to Poisson, albeit providing a deterministic representation of probabilities. We proceed to develop a theory of Bayesian inference in spiking neural networks, recurrent interactions implementing a variant of belief propagation. Many perceptual and motor tasks performed by the central nervous system are probabilistic, and can be described in a Bayesian framework [4, 3]. A few important but hidden properties, such as direction of motion, or appropriate motor commands, are inferred from many noisy, local and ambiguous sensory cues. These evidences are combined with priors about the sensory world and body. Importantly, because most of these inferences should lead to quick and irreversible decisions in a perpetually changing world, noisy cues have to be integrated on-line, but in a way that takes into account unpredictable events, such as a sudden change in motion direction or the appearance of a new stimulus. This raises the question of how this temporal integration can be performed at the neural level. It has been proposed that single neurons in sensory cortices represent and compute the log probability that a sensory variable takes on a certain value (eg Is visual motion in the neuron’s preferred direction?) [9, 7]. Alternatively, to avoid normalization issues and provide an appropriate signal for decision making, neurons could represent the log probability ratio of a particular hypothesis (eg is motion more likely to be towards the right than towards the left) [7, 6]. Log probabilities are convenient here, since under some assumptions, independent noisy cues simply combine linearly. Moreover, there are physiological evidence for the neural representation of log probabilities and log probability ratios [9, 6, 7]. However, these models assume that neurons represent probabilities in their firing rates. We argue that it is important to study how probabilistic information are encoded in spikes. Indeed, it seems spurious to marry the idea of an exquisite on-line integration of noisy cues with an underlying rate code that requires averaging on large populations of noisy neurons and long periods of time. In particular, most natural tasks require this integration to take place on the time scale of inter-spike intervals. Spikes are more efficiently signaling events ∗ Institute of Cognitive Science, 69645 Bron, France than analog quantities. In addition, a neural theory of inference with spikes will bring us closer to the physiological level and generate more easily testable predictions. Thus, we propose a new theory of neural processing in which spike trains provide a deterministic, online representation of a log-probability ratio. Spikes signals events, eg that the log-probability ratio has exceeded what could be predicted from previous spikes. This form of coding was loosely inspired by the idea of ”energy landscape” coding proposed by Hinton and Brown [2]. However, contrary to [2] and other theories using rate-based representation of probabilities, this model is self-consistent and does not require different models for encoding and decoding: As output spikes provide new, unpredictable, temporally independent evidence, they can be used directly as an input to other Bayesian neurons. Finally, we show that these neurons can be used as building blocks in a theory of approximate Bayesian inference in recurrent spiking networks. Connections between neurons implement an underlying Bayesian network, consisting of coupled hidden Markov models. Propagation of spikes is a form of belief propagation in this underlying graphical model. Our theory provides computational explanations of some general physiological properties of cortical neurons, such as spike frequency adaptation, Poisson statistics of spike trains, the existence of strong local inhibition in cortical columns, and the maintenance of a tight balance between excitation and inhibition. Finally, we discuss the implications of this model for the debate about temporal versus rate-based neural coding. 1 Spikes and log posterior odds 1.1 Synaptic integration seen as inference in a hidden Markov chain We propose that each neuron codes for an underlying ”hidden” binary variable, xt , whose state evolves over time. We assume that xt depends only on the state at the previous time step, xt−dt , and is conditionally independent of other past states. The state xt can switch 1 from 0 to 1 with a constant rate ron = dt limdt→0 P (xt = 1|xt−dt = 0), and from 1 to 0 with a constant rate roff . For example, these transition rates could represent how often motion in a preferred direction appears the receptive field and how long it is likely to stay there. The neuron infers the state of its hidden variable from N noisy synaptic inputs, considered to be observations of the hidden state. In this initial version of the model, we assume that these inputs are conditionally independent homogeneous Poisson processes, synapse i i emitting a spike between time t and t + dt (si = 1) with constant probability qon dt if t i xt = 1, and another constant probability qoff dt if xt = 0. The synaptic spikes are assumed to be otherwise independent of previous synaptic spikes, previous states and spikes at other synapses. The resulting generative model is a hidden Markov chain (figure 1-A). However, rather than estimating the state of its hidden variable and communicating this estimate to other neurons (for example by emitting a spike when sensory evidence for xt = 1 goes above a threshold) the neuron reports and communicates its certainty that the current state is 1. This certainty takes the form of the log of the ratio of the probability that the hidden state is 1, and the probability that the state is 0, given all the synaptic inputs P (xt =1|s0→t ) received so far: Lt = log P (xt =0|s0→t ) . We use s0→t as a short hand notation for the N synaptic inputs received at present and in the past. We will refer to it as the log odds ratio. Thanks to the conditional independencies assumed in the generative model, we can compute this Log odds ratio iteratively. Taking the limit as dt goes to zero, we get the following differential equation: ˙ L = ron 1 + e−L − roff 1 + eL + i wi δ(si − 1) − θ t B. A. xt ron .roff dt qon , qoff st xt ron .roff i t st dt s qon , qoff qon , qoff st dt xt j st Ot It Gt Ot Lt t t dt C. E. 2 0 -2 -4 D. 500 1000 1500 2000 2500 2 3000 Count Log odds 4 20 Lt 0 -2 0 500 1000 1500 2000 2500 Time Ot 3000 0 200 400 600 ISI Figure 1: A. Generative model for the synaptic input. B. Schematic representation of log odds ratio encoding and decoding. The dashed circle represents both eventual downstream elements and the self-prediction taking place inside the model neuron. A spike is fired only when Lt exceeds Gt . C. One example trial, where the state switches from 0 to 1 (shaded area) and back to 0. plain: Lt , dotted: Gt . Black stripes at the top: corresponding spikes train. D. Mean Log odds ratio (dark line) and mean output firing rate (clear line). E. Output spike raster plot (1 line per trial) and ISI distribution for the neuron shown is C. and D. Clear line: ISI distribution for a poisson neuron with the same rate. wi , the synaptic weight, describe how informative synapse i is about the state of the hidden i qon variable, e.g. wi = log qi . Each synaptic spike (si = 1) gives an impulse to the log t off odds ratio, which is positive if this synapse is more active when the hidden state if 1 (i.e it increases the neuron’s confidence that the state is 1), and negative if this synapse is more active when xt = 0 (i.e it decreases the neuron’s confidence that the state is 1). The bias, θ, is determined by how informative it is not to receive any spike, e.g. θ = i i i qon − qoff . By convention, we will consider that the ”bias” is positive or zero (if not, we need simply to invert the status of the state x). 1.2 Generation of output spikes The spike train should convey a sparse representation of Lt , so that each spike reports new information about the state xt that is not redundant with that reported by other, preceding, spikes. This proposition is based on three arguments: First, spikes, being metabolically expensive, should be kept to a minimum. Second, spikes conveying redundant information would require a decoding of the entire spike train, whereas independent spike can be taken into account individually. And finally, we seek a self consistent model, with the spiking output having a similar semantics to its spiking input. To maximize the independence of the spikes (conditioned on xt ), we propose that the neuron fires only when the difference between its log odds ratio Lt and a prediction Gt of this log odds ratio based on the output spikes emitted so far reaches a certain threshold. Indeed, supposing that downstream elements predicts Lt as best as they can, the neuron only needs to fire when it expects that prediction to be too inaccurate (figure 1-B). In practice, this will happen when the neuron receives new evidence for xt = 1. Gt should thereby follow the same dynamics as Lt when spikes are not received. The equation for Gt and the output Ot (Ot = 1 when an output spike is fired) are given by: ˙ G = Ot = ron 1 + e−L − roff 1 + eL + go δ(Ot − 1) go 1. when Lt > Gt + , 0 otherwise, 2 (1) (2) Here go , a positive constant, is the only free parameter, the other parameters being constrained by the statistics of the synaptic input. 1.3 Results Figure 1-C plots a typical trial, showing the behavior of L, G and O before, during and after presentation of the stimulus. As random synaptic inputs are integrated, L fluctuates and eventually exceeds G + 0.5, leading to an output spike. Immediately after a spike, G jumps to G + go , which prevents (except in very rare cases) a second spike from immediately following the first. Thus, this ”jump” implements a relative refractory period. However, ron G decays as it tends to converge back to its stable level gstable = log roff . Thus L eventually exceeds G again, leading to a new spike. This threshold crossing happens more often during stimulation (xt = 1) as the net synaptic input alters to create a higher overall level of certainty, Lt . Mean Log odds ratio and output firing rate ¯ The mean firing rate Ot of the Bayesian neuron during presentation of its preferred stimulus (i.e. when xt switches from 0 to 1 and back to 0) is plotted in figure 1-D, together with the ¯ mean log posterior ratio Lt , both averaged over trials. Not surprisingly, the log-posterior ratio reflects the leaky integration of synaptic evidence, with an effective time constant that depends on the transition probabilities ron , roff . If the state is very stable (ron = roff ∼ 0), synaptic evidence is integrated over almost infinite time periods, the mean log posterior ratio tending to either increase or decrease linearly with time. In the example in figure 1D, the state is less stable, so ”old” synaptic evidence are discounted and Lt saturates. ¯ In contrast, the mean output firing rate Ot tracks the state of xt almost perfectly. This is because, as a form of predictive coding, the output spikes reflect the new synaptic i evidence, It = i δ(st − 1) − θ, rather than the log posterior ratio itself. In particular, the mean output firing rate is a rectified linear function of the mean input, e. g. + ¯ ¯ wi q i −θ . O= 1I= go i on(off) Analogy with a leaky integrate and fire neuron We can get an interesting insight into the computation performed by this neuron by linearizing L and G around their mean levels over trials. Here we reduce the analysis to prolonged, statistically stable periods when the state is constant (either ON or OFF). In this case, the ¯ ¯ mean level of certainty L and its output prediction G are also constant over time. We make the rough approximation that the post spike jump, go , and the input fluctuations are small ¯ compared to the mean level of certainty L. Rewriting Vt = Lt − Gt + go 2 as the ”membrane potential” of the Bayesian neuron: ˙ V = −kL V + It − ∆go − go Ot ¯ ¯ ¯ where kL = ron e−L + roff eL , the ”leak” of the membrane potential, depends on the overall ¯ level of certainty. ∆go is positive and a monotonic increasing function of go . A. s t1 dt s t1 s t1 dt B. C. x t1 x t3 dt x t3 x t3 dt x t1 x t1 x t1 x t2 x t3 x t1 … x tn x t3 x t2 … x tn … dt dt Lx2 D. x t2 dt s t2 dt x t2 s t2 x t2 dt s t2 dt Log odds 10 No inh -0.5 -1 -1 -1.5 -2 5 Feedback 500 1000 1500 2000 Tiger Stripes 0 -5 -10 500 1000 1500 2000 2500 Time Figure 2: A. Bayesian causal network for yt (tiger), x1 (stripes) and x2 (paws). B. A nett t work feedforward computing the log posterior for x1 . C. A recurrent network computing t the log posterior odds for all variables. D. Log odds ratio in a simulated trial with the net2 1 1 work in C (see text). Thick line: Lx , thin line: Lx , dash-dotted: Lx without inhibition. t t t 2 Insert: Lx averaged over trials, showing the effect of feedback. t The linearized Bayesian neuron thus acts in its stable regime as a leaky integrate and fire (LIF) neuron. The membrane potential Vt integrates its input, Jt = It − ∆go , with a leak kL . The neuron fires when its membrane potential reaches a constant threshold go . After ¯ each spikes, Vt is reset to 0. Interestingly, for appropriately chosen compression factor go , the mean input to the lin¯ ¯ earized neuron J = I − ∆go ≈ 0 1 . This means that the membrane potential is purely driven to its threshold by input fluctuations, or a random walk in membrane potential. As a consequence, the neuron’s firing will be memoryless, and close to a Poisson process. In particular, we found Fano factor close to 1 and quasi-exponential ISI distribution (figure 1E) on the entire range of parameters tested. Indeed, LIF neurons with balanced inputs have been proposed as a model to reproduce the statistics of real cortical neurons [8]. This balance is implemented in our model by the neuron’s effective self-inhibition, even when the synaptic input itself is not balanced. Decoding As we previously said, downstream elements could predict the log odds ratio Lt by computing Gt from the output spikes (Eq 1, fig 1-B). Of course, this requires an estimate of the transition probabilities ron , roff , that could be learned from the observed spike trains. However, we show next that explicit decoding is not necessary to perform bayesian inference in spiking networks. Intuitively, this is because the quantity that our model neurons receive and transmit, eg new information, is exactly what probabilistic inference algorithm propagate between connected statistical elements. 1 ¯ Even if go is not chosen optimally, the influence of the drift J is usually negligible compared to the large fluctuations in membrane potential. 2 Bayesian inference in cortical networks The model neurons, having the same input and output semantics, can be used as building blocks to implement more complex generative models consisting of coupled Markov chains. Consider, for example, the example in figure 2-A. Here, a ”parent” variable x1 t (the presence of a tiger) can cause the state of n other ”children” variables ([xk ]k=2...n ), t of whom two are represented (the presence of stripes,x2 , and motion, x3 ). The ”chilt t dren” variables are Bayesian neurons identical to those described previously. The resulting bayesian network consist of n + 1 coupled hidden Markov chains. Inference in this architecture corresponds to computing the log posterior odds ratio for the tiger, x1 , and the log t posterior of observing stripes or motion, ([xk ]k=2...n ), given the synaptic inputs received t by the entire network so far, i.e. s2 , . . . , sk . 0→t 0→t Unfortunately, inference and learning in this network (and in general in coupled Markov chains) requires very expensive computations, and cannot be performed by simply propagating messages over time and among the variable nodes. In particular, the state of a child k variable xt depends on xk , sk , x1 and the state of all other children at the previous t t t−dt time step, [xj ]2