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76 nips-2004-Hierarchical Bayesian Inference in Networks of Spiking Neurons

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Author: Rajesh P. Rao

Abstract: There is growing evidence from psychophysical and neurophysiological studies that the brain utilizes Bayesian principles for inference and decision making. An important open question is how Bayesian inference for arbitrary graphical models can be implemented in networks of spiking neurons. In this paper, we show that recurrent networks of noisy integrate-and-ﬁre neurons can perform approximate Bayesian inference for dynamic and hierarchical graphical models. The membrane potential dynamics of neurons is used to implement belief propagation in the log domain. The spiking probability of a neuron is shown to approximate the posterior probability of the preferred state encoded by the neuron, given past inputs. We illustrate the model using two examples: (1) a motion detection network in which the spiking probability of a direction-selective neuron becomes proportional to the posterior probability of motion in a preferred direction, and (2) a two-level hierarchical network that produces attentional effects similar to those observed in visual cortical areas V2 and V4. The hierarchical model offers a new Bayesian interpretation of attentional modulation in V2 and V4. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract There is growing evidence from psychophysical and neurophysiological studies that the brain utilizes Bayesian principles for inference and decision making. [sent-5, score-0.179]

2 An important open question is how Bayesian inference for arbitrary graphical models can be implemented in networks of spiking neurons. [sent-6, score-0.591]

3 In this paper, we show that recurrent networks of noisy integrate-and-ﬁre neurons can perform approximate Bayesian inference for dynamic and hierarchical graphical models. [sent-7, score-1.126]

4 The membrane potential dynamics of neurons is used to implement belief propagation in the log domain. [sent-8, score-0.824]

5 The spiking probability of a neuron is shown to approximate the posterior probability of the preferred state encoded by the neuron, given past inputs. [sent-9, score-1.069]

6 The hierarchical model offers a new Bayesian interpretation of attentional modulation in V2 and V4. [sent-11, score-0.223]

7 1 Introduction A wide range of psychophysical results have recently been successfully explained using Bayesian models [7, 8, 16, 19]. [sent-12, score-0.032]

8 These models have been able to account for human responses in tasks ranging from 3D shape perception to visuomotor control. [sent-13, score-0.125]

9 Simultaneously, there is accumulating evidence from human and monkey experiments that Bayesian mechanisms are at work during visual decision making [2, 5]. [sent-14, score-0.051]

10 An important question that has only recently received attention is how networks of cortical neurons can implement algorithms for Bayesian inference. [sent-16, score-0.598]

11 One powerful approach has been to build on the known properties of population coding models that represent information using a set of neural tuning curves or kernel functions [1, 20]. [sent-17, score-0.068]

12 Several proposals have been made regarding how a probability distribution could be encoded using population codes ([3, 18]; see [14] for an excellent review). [sent-18, score-0.209]

13 However, the problem of implementing general inference algorithms for arbitrary graphical models using population codes remains unresolved (some encouraging initial results are reported in Zemel et al. [sent-19, score-0.384]

14 For example, a neural implementation of approximate Bayesian inference for a hidden Markov model was investigated in [15]. [sent-22, score-0.15]

15 The question of how such an approach could be generalized to spiking neurons and arbitrary graphical models remained open. [sent-23, score-0.747]

16 In this paper, we propose a method for implementing Bayesian belief propagation in networks of spiking neurons. [sent-24, score-0.588]

17 We show that recurrent networks of noisy integrate-and-ﬁre neurons can perform approximate Bayesian inference for dynamic and hierarchical graphical models. [sent-25, score-1.126]

18 In the model, the dynamics of the membrane potential is used to implement on-line belief propagation in the log domain [15]. [sent-26, score-0.503]

19 A neuron’s spiking probability is shown to approximate the posterior probability of the preferred state encoded by the neuron, given past inputs. [sent-27, score-0.72]

20 We ﬁrst show that for a visual motion detection task, the spiking probability of a direction-selective neuron becomes proportional to the posterior probability of motion in the neuron’s preferred direction. [sent-28, score-1.384]

21 We then show that in a two-level network, hierarchical Bayesian inference [9] produces responses that mimic the attentional effects seen in visual cortical areas V2 and V4. [sent-29, score-0.657]

22 1 Integrate-and-Fire Model of Spiking Neurons We begin with a recurrently-connected network of integrate-and-ﬁre (IF) neurons receiving feedforward inputs denoted by the vector I. [sent-31, score-0.569]

23 If vi crosses a threshold T , the neuron ﬁres a spike and vi is reset to the potential vreset . [sent-33, score-0.684]

24 Equation 1 can be rewritten in discrete form as: vi (t + 1) = vi (t) + (−vi (t) + wij Ij (t)) + j i. [sent-34, score-0.334]

25 vi (t + 1) uij vj (t) wij Ij (t) + = uij vj (t)) (2) j (3) j j where is the integration rate, uii = 1 + (uii − 1) and for i = j, uij = uij . [sent-36, score-1.091]

26 2 Stochastic Spiking in Noisy IF Neurons To model the effects of background inputs and the random openings of membrane channels, one can add a Gaussian white noise term to the right hand side of Equations 3 and 4. [sent-39, score-0.272]

27 This makes the spiking of neurons in the recurrent network stochastic. [sent-40, score-0.928]

28 Of particular interest to the present paper is the following exponential function for spiking probability suggested in [4] for noisy integrate-and-ﬁre networks: P (neuron i spikes at time t) = ke(vi (t)−T )/c (5) where k and c are arbitrary constants. [sent-43, score-0.399]

29 1 Inference in a Single-Level Model We ﬁrst consider on-line belief propagation in a single-level dynamic graphical model and show how it can be implemented in spiking networks. [sent-46, score-0.648]

30 The graphical model is shown in Figure 1A and corresponds to a classical hidden Markov model. [sent-47, score-0.117]

31 Let θ(t) represent the hidden state of a Markov model at time t with transition probabilities given by P (θ(t) = t t−1 θi |θ(t − 1) = θj ) = P (θi |θj ) for i, j = 1 . [sent-48, score-0.199]

32 Let I(t) be the observable output governed by the probabilities P (I(t)|θ(t)). [sent-52, score-0.07]

33 2 Neural Implementation of the Inference Algorithm By comparing the membrane potential equation (Eq. [sent-61, score-0.229]

34 7), it is clear that the ﬁrst equation can implement the second if belief propagation is performed in the log domain [15], i. [sent-63, score-0.356]

35 Normalization by nt−1,t is implemented by subtracting log nt−1,t using inhibition. [sent-66, score-0.035]

36 1 An alternative approach, which was also found to yield satisfactory results, is to approximate the log-sum with a linear weighted sum [15], the weights being chosen to minimize the approximation error. [sent-67, score-0.07]

37 Each circle represents a node denoting the state variable θ t which can take on values θ1 , . [sent-70, score-0.071]

38 (B) Recurrent network for implementing on-line belief propagation for the graphical model in (A). [sent-74, score-0.456]

39 Each circle represents a neuron encoding a state θ i . [sent-75, score-0.42]

40 The probability distribution over state values at each time step is represented by the entire population. [sent-77, score-0.103]

41 (D) Two-level network for implementing online belief propagation for the graphical model in (C). [sent-79, score-0.456]

42 Arrows represent synaptic connections in the direction pointed by the arrow heads. [sent-80, score-0.234]

43 Finally, since the membrane potential vi (t + 1) is assumed to be proportional to log mt,t+1 i (Equation 9), we have: vi (t + 1) = c log mt,t+1 + T (12) i for some constants c and T . [sent-82, score-0.524]

44 Figure 1B illustrates the singlelevel recurrent network model that implements the on-line belief propagation equation 7. [sent-84, score-0.617]

45 3 Hierarchical Inference The model described above can be extended to perform on-line belief propagation and inference for arbitrary graphical models. [sent-86, score-0.449]

46 As an example, we describe the implementation for the two-level hierarchical graphical model in Figure 1C. [sent-87, score-0.243]

47 The equations above can be implemented in a 2-level hierarchical recurrent network of integrate-and-ﬁre neurons in a manner similar to the 1-level case. [sent-90, score-0.818]

48 We assume that neuron i in level 1 encodes θ1,i as its preferred state while neuron i in level 2 encodes θ2,i . [sent-91, score-1.029]

49 We also assume speciﬁc feedforward and feedback neurons for computing and t conveying mt 1→2,i and m2→1,i respectively. [sent-92, score-0.703]

50 Taking the logarithm of both sides of Equations 17 and 18, we obtain equations that can be computed using the membrane potential dynamics of integrate-and-ﬁre neurons (Equation 4). [sent-93, score-0.565]

51 , [10]) and potential implementations based on active dendritic interactions have been explored. [sent-97, score-0.195]

52 The model suggested here utilizes these multiplicative interactions within dendritic branches, in addition to a possible logarithmic transform of the signal before it sums with other signals at the soma. [sent-98, score-0.178]

53 Such a model is comparable to recent models of dendritic computation (see [6] for more details). [sent-99, score-0.102]

54 1 Single-Level Network: Probabilistic Motion Detection and Direction Selectivity We ﬁrst tested the model in a 1D visual motion detection task [15]. [sent-101, score-0.295]

55 A single-level recurrent network of 30 neurons was used (see Figure 1B). [sent-102, score-0.658]

56 Figure 2A shows the feedforward weights for neurons 1, . [sent-103, score-0.5]

57 , 15: these were recurrently connected to encode transition probabilities biased for rightward motion as shown in Figure 2B. [sent-106, score-0.484]

58 , 30 were identical to Figure 2A but their recurrent connections encoded transition probabilities for leftward motion (see Figure 2B). [sent-110, score-0.872]

59 As seen in Figure 2C, neurons in the network exhibited direction selectivity. [sent-111, score-0.437]

60 Furthermore, the spiking probability of neurons reﬂected the posterior probabilities over time of motion direction at a given location (Figure 2D), suggesting a probabilistic interpretation of direction selective spiking responses in visual cortical areas such as V1 and MT. [sent-112, score-1.66]

61 2 Two-Level Network: Spatial Attention as Hierarchical Bayesian Inference We tested the two-level network implementation (Figure 1D) of hierarchical Bayesian inference using a simple attention task previously used in primate studies [17]. [sent-114, score-0.431]

62 In an input image, a vertical or horizontal bar could occur either on the left side, right side, or both sides (see Figure 3). [sent-115, score-0.183]

63 The corresponding 2-level generative model consisted of two states at level 2 (left or right side) and four states at level 1: vertical left, horizontal left, vertical right, horizontal right. [sent-116, score-0.489]

64 Each of these states was encoded by a neuron at the respective level. [sent-117, score-0.469]

65 The feedforward connections at level 1 were chosen to be vertically or horizontally oriented Gabor ﬁlters localized to the left or right side of the image. [sent-118, score-0.36]

66 Since the experiment used static images, the recurrent connections at each level implemented transition probabilities close to 1 for the same state and small random values for other states. [sent-119, score-0.598]

67 The transition probabilities t−1 t t t P (θ1,k |θ2,i , θ1,j ) were chosen such that for θ2 = left side, the transition probabilities for θt w15 w1 1 0. [sent-120, score-0.256]

68 (B) Recurrent weights encoding the transition t+1 t probabilities P (θi |θj ) for i, j = 1, . [sent-165, score-0.163]

69 (C) Spiking responses of three of the ﬁrst 15 neurons in the recurrent network (neurons 8, 10, and 12). [sent-170, score-0.75]

70 As is evident, these neurons have become selective for rightward motion as a consequence of the recurrent connections speciﬁed in (B). [sent-171, score-1.038]

71 (D) Posterior probabilities over time of motion direction (at a given location) encoded by the three neurons for rightward and leftward motion. [sent-172, score-1.015]

72 t t states θ1 coding for the right side were set to values close to zero (and vice versa, for θ 2 = right side). [sent-173, score-0.158]

73 As shown in Figure 3, the response of a neuron at level 1 that, for example, prefers a vertical edge on the right mimics the response of a V4 neuron with and without attention (see ﬁgure caption for more details). [sent-174, score-0.943]

74 The initial setting of the priors at level 2 is the crucial determinant of attentional modulation in level 1 neurons, suggesting that feedback from higher cortical areas may convey task-speciﬁc priors that are integrated into V4 responses. [sent-175, score-0.441]

75 5 Discussion and Conclusions We have shown that recurrent networks of noisy integrate-and-ﬁre neurons can perform approximate Bayesian inference for single- and multi-level dynamic graphical models. [sent-176, score-1.0]

76 The model suggests a new interpretation of the spiking probability of a neuron in terms of the posterior probability of the preferred state encoded by the neuron, given past inputs. [sent-177, score-1.034]

77 We illustrated the model using two problems: inference of motion direction in a single-level network and hierarchical inference of object identity at an attended visual location in a twolevel network. [sent-178, score-0.73]

78 In the ﬁrst case, neurons generated direction-selective spikes encoding the probability of motion in a particular direction. [sent-179, score-0.605]

79 In the second case, attentional effects similar to those observed in primate cortical areas V2 and V4 emerged as a result of imposing appropriate priors at the highest level. [sent-180, score-0.331]

80 How does the approach scale to more realistic graphical models? [sent-182, score-0.117]

81 The two-level model explored in this paper assumed stationary objects, resulting in simpliﬁed dynamics for the two levels in our recurrent network. [sent-183, score-0.31]

82 Experiments are currently underway to test the robustness of the proposed model when richer classes of dynamics are introduced at the different levels. [sent-184, score-0.044]

83 (A) Top panel: Input image (lasting the ﬁrst 15 time steps) containing a vertical bar (“Reference”) on the right side. [sent-186, score-0.151]

84 Middle: Three sample spike trains from the 1st level neuron whose preferred stimulus was a vertical bar on the right side. [sent-188, score-0.709]

85 Bottom: Posterior probability of a vertical bar (= spiking probability or instantaneous ﬁring rate of the neuron) plotted over time. [sent-189, score-0.493]

86 Below: Sample spike trains and posterior probability for the same neuron as in (A). [sent-191, score-0.488]

87 (D) Responses from a neuron in primate area V4 without attention (top panel, Ref Att Away and Pair Att Away; compare with (A) and (B)) and with attention (bottom panel, Pair Att Ref; compare with (C)) (from [17]). [sent-193, score-0.529]

88 other open question is how active dendritic processes could support probabilistic integration of messages from local, lower-level, and higher-level neurons, as suggested in Section 3. [sent-195, score-0.18]

89 We intend to investigate this question using biophysical (compartmental) models of cortical neurons. [sent-196, score-0.125]

90 Finally, how can the feedforward, feedback, and recurrent synaptic weights in the networks be learned directly from input data? [sent-197, score-0.46]

91 We hope to investigate this question using biologically-plausible approximations to the expectation-maximization (EM) algorithm. [sent-198, score-0.039]

92 Neural computation of log likelihood in control of saccadic eye movements. [sent-217, score-0.035]

93 Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. [sent-228, score-0.314]

94 Loopy belief propagation for approximate inference: An empirical study. [sent-269, score-0.252]

95 Noise in integrate-and-ﬁre neurons: From stochastic input to escape rates. [sent-280, score-0.045]

96 Competitive mechanisms subserve attention in macaque areas V2 and V4. [sent-310, score-0.112]

97 Doubly distributional population codes: Simultaneous representation of uncertainty and multiplicity. [sent-315, score-0.068]

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same-paper 1 0.99999934 76 nips-2004-Hierarchical Bayesian Inference in Networks of Spiking Neurons

Author: Rajesh P. Rao

2 0.34397829 28 nips-2004-Bayesian inference in spiking neurons

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. 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