BACKGROUND: Mood disorders involve a complex interplay between multifaceted internal emotional states, and complex external inputs. Dynamical systems theory suggests that this interplay between aspects of moods and environmental stimuli may hence determine key psychopathological features of mood disorders, including the stability of mood states, the response to external inputs, how controllable mood states are, and what interventions are most likely to be effective. However, a comprehensive computational approach to all these aspects has not yet been undertaken. METHODS: Here, we argue that the combination of ecological momentary assessments (EMA) with a well-established dynamical systems framework-the humble Kalman filter-enables a comprehensive account of all these aspects. We first introduce the key features of the Kalman filter and optimal control theory and their relationship to aspects of psychopathology. We then examine the psychometric and inferential properties of combining EMA data with Kalman filtering across realistic scenarios. Finally, we apply the Kalman filter to a series of EMA datasets comprising over 700 participants with and without symptoms of depression. RESULTS: The results show a naive Kalman filter approach performs favourably compared to the standard vector autoregressive approach frequently employed, capturing key aspects of the data better. Furthermore, it suggests that the depressed state involves alterations to interactions between moods; alterations to how moods responds to external inputs; and as a result an alteration in how controllable mood states are. We replicate these findings qualitatively across datasets and explore an extension to optimal control theory to guide therapeutic interventions. CONCLUSIONS: Mood dynamics are richly and profoundly altered in depressed states. The humble Kalman filter is a well-established, rich framework to characterise mood dynamics. Its application to EMA data is valid; straightforward; and likely to result in substantial novel insights both into mechanisms and treatments.

Metacognition is the ability to reflect on, and evaluate, our cognition and behaviour. Distortions in metacognition are common in mental health disorders, though the neural underpinnings of such dysfunction are unknown. One reason for this is that models of key components of metacognition, such as decision confidence, are generally specified at an algorithmic or process level. While such models can be used to relate brain function to psychopathology, they are difficult to map to a neurobiological mechanism. Here, we develop a biologically-plausible model of decision uncertainty in an attempt to bridge this gap. We first relate the model's uncertainty in perceptual decisions to standard metrics of metacognition, namely mean confidence level (bias) and the accuracy of metacognitive judgments (sensitivity). We show that dissociable shifts in metacognition are associated with isolated disturbances at higher-order levels of a circuit associated with self-monitoring, akin to neuropsychological findings that highlight the detrimental effect of prefrontal brain lesions on metacognitive performance. Notably, we are able to account for empirical confidence judgements by fitting the parameters of our biophysical model to first-order performance data, specifically choice and response times. Lastly, in a reanalysis of existing data we show that self-reported mental health symptoms relate to disturbances in an uncertainty-monitoring component of the network. By bridging a gap between a biologically-plausible model of confidence formation and observed disturbances of metacognition in mental health disorders we provide a first step towards mapping theoretical constructs of metacognition onto dynamical models of decision uncertainty. In doing so, we provide a computational framework for modelling metacognitive performance in settings where access to explicit confidence reports is not possible.

Human decisions can be habitual or goal-directed, also known as model-free (MF) or model-based (MB) control.Previous work suggests that the balance between the two decision systems is impaired in psychiatric disorders such as compulsion and addiction, via overreliance on MF control.However, little is known whether the balance can be altered through task training.Here, 20 healthy participants performed a well-established two-step task that differentiates MB from MF control, across five training sessions.We used computational modelling and functional near-infrared spectroscopy to assess changes in decisionmaking and brain hemodynamic over time.Mixed-effects modelling revealed overall no substantial changes in MF and MB behavior across training.Although our behavioral and brain findings show task-induced changes in learning rates, these parameters have no direct relation to either MF or MB control or the balance between the two systems, and thus do not support the assumption of training effects on MF or MB strategies.Our findings indicate that training on the two-step paradigm in its current form does not support a shift in the balance between MF and MB control.We discuss these results with respect to implications for restoring the balance between MF and MB control in psychiatric conditions. Author summaryPsychiatric conditions such as compulsion or addiction are associated with an overreliance on habitual, or model-free, decision-making.Goal-directed, or model-based, decision-making may protect against such overreliance.We therefore asked whether modelfree control could be reduced, and model-based control strengthened, via task training.We used the well-characterized two-step task that differentiates model-based from modelfree actions.Our results suggest that training on the current form of the two-step task does not support a shift in the balance between model-free and model-based strategies.Factors such as devaluation, demotivation or automatization during training may play a

Optimists hold positive a priori beliefs about the future. In Bayesian statistical theory, a priori beliefs can be overcome by experience. However, optimistic beliefs can at times appear surprisingly resistant to evidence, suggesting that optimism might also influence how new information is selected and learned. Here, we use a novel Pavlovian conditioning task, embedded in a normative framework, to directly assess how trait optimism, as classically measured using self-report questionnaires, influences choices between visual targets, by learning about their association with reward progresses. We find that trait optimism relates to an a priori belief about the likelihood of rewards, but not losses, in our task. Critically, this positive belief behaves like a probabilistic prior, i.e. its influence reduces with increasing experience. Contrary to findings in the literature related to unrealistic optimism and self-beliefs, it does not appear to influence the iterative learning process directly.

When planning a series of actions, it is usually infeasible to consider all potential future sequences; instead, one must prune the decision tree. Provably optimal pruning is, however, still computationally ruinous and the specific approximations humans employ remain unknown. We designed a new sequential reinforcement-based task and showed that human subjects adopted a simple pruning strategy: during mental evaluation of a sequence of choices, they curtailed any further evaluation of a sequence as soon as they encountered a large loss. This pruning strategy was Pavlovian: it was reflexively evoked by large losses and persisted even when overwhelmingly counterproductive. It was also evident above and beyond loss aversion. We found that the tendency towards Pavlovian pruning was selectively predicted by the degree to which subjects exhibited sub-clinical mood disturbance, in accordance with theories that ascribe Pavlovian behavioural inhibition, via serotonin, a role in mood disorders. We conclude that Pavlovian behavioural inhibition shapes highly flexible, goal-directed choices in a manner that may be important for theories of decision-making in mood disorders.

Hard-wired, Pavlovian, responses elicited by predictions of rewards and punishments exert significant benevolent and malevolent influences over instrumentally-appropriate actions. These influences come in two main groups, defined along anatomical, pharmacological, behavioural and functional lines. Investigations of the influences have so far concentrated on the groups as a whole; here we take the critical step of looking inside each group, using a detailed reinforcement learning model to distinguish effects to do with value, specific actions, and general activation or inhibition. We show a high degree of sophistication in Pavlovian influences, with appetitive Pavlovian stimuli specifically promoting approach and inhibiting withdrawal, and aversive Pavlovian stimuli promoting withdrawal and inhibiting approach. These influences account for differences in the instrumental performance of approach and withdrawal behaviours. Finally, although losses are as informative as gains, we find that subjects neglect losses in their instrumental learning. Our findings argue for a view of the Pavlovian system as a constraint or prior, facilitating learning by alleviating computational costs that come with increased flexibility.

Biophysically detailed models of single cells are difficult to fit to real data. Recent advances in imaging techniques allow simultaneous access to various intracellular variables, and these data can be used to significantly facilitate the modelling task. These data, however, are noisy, and current approaches to building biophysically detailed models are not designed to deal with this. We extend previous techniques to take the noisy nature of the measurements into account. Sequential Monte Carlo ("particle filtering") methods, in combination with a detailed biophysical description of a cell, are used for principled, model-based smoothing of noisy recording data. We also provide an alternative formulation of smoothing where the neural nonlinearities are estimated in a non-parametric manner. Biophysically important parameters of detailed models (such as channel densities, intercompartmental conductances, input resistances, and observation noise) are inferred automatically from noisy data via expectation-maximization. Overall, we find that model-based smoothing is a powerful, robust technique for smoothing of noisy biophysical data and for inference of biophysical parameters in the face of recording noise.

Decision making lies at the very heart of many psychiatric diseases. It is also a central theoretical concern in a wide variety of fields and has undergone detailed, in-depth, analyses. We take as an example Major Depressive Disorder (MDD), applying insights from a Bayesian reinforcement learning framework. We focus on anhedonia and helplessness. Helplessness—a core element in the conceptual- izations of MDD that has lead to major advances in its treatment, pharmacolog- ical and neurobiological understanding—is formalized as a simple prior over the outcome entropy of actions in uncertain environments. Anhedonia, which is an equally fundamental aspect of the disease, is related to the effective reward size. These formulations allow for the design of specific tasks to measure anhedonia and helplessness behaviorally. We show that these behavioral measures capture explicit, questionnaire-based cognitions. We also provide evidence that these tasks may allow classification of subjects into healthy and MDD groups based purely on a behavioural measure and avoiding any verbal reports. There are strong ties between decision making and psychiatry, with maladaptive decisions and be- haviors being very prominent in people with psychiatric disorders. Depression is classically seen as following life events such as divorces and job losses. Longitudinal studies, however, have revealed that a significant fraction of the stressors associated with depression do in fact follow MDD onset, and that they are likely due to maladaptive behaviors prominent in MDD (Kendler et al., 1999). Clinically effective ’talking’ therapies for MDD such as cognitive and dialectical behavior therapies (DeRubeis et al., 1999; Bortolotti et al., 2008; Gotlib and Hammen, 2002; Power, 2005) explicitly concentrate on altering patients’ maladaptive behaviors and decision making processes. Decision making is a promising avenue into psychiatry for at least two more reasons. First, it offers powerful analytical tools. Control problems related to decision making are prevalent in a huge diversity of fields, ranging from ecology to economics, computer science and engineering. These fields have produced well-founded and thoroughly characterized frameworks within which many issues in decision making can be framed. Here, we will focus on framing issues identified in psychiatric settings within a normative decision making framework. Its second major strength comes from its relationship to neurobiology, and particularly those neuro- modulatory systems which are powerfully affected by all major clinically effective pharmacothera- pies in psychiatry. The understanding of these systems has benefited significantly from theoretical accounts of optimal control such as reinforcement learning (Montague et al., 1996; Kapur and Rem- ington, 1996; Smith et al., 1999; Yu and Dayan, 2005; Dayan and Yu, 2006). Such accounts may be useful to identify in more specific terms the roles of the neuromodulators in psychiatry (Smith et al., 2004; Williams and Dayan, 2005; Moutoussis et al., 2008; Dayan and Huys, 2008). ∗[email protected], [email protected], [email protected]; www.gatsby.ucl.ac.uk/∼qhuys/pub.html

Naturally occurring sensory stimuli are dynamic. In this letter, we consider how spiking neural populations might transmit information about continuous dynamic stimulus variables. The combination of simple encoders and temporal stimulus correlations leads to a code in which information is not readily available to downstream neurons. Here, we explore a complex encoder that is paired with a simple decoder that allows representation and manipulation of the dynamic information in neural systems. The encoder we present takes the form of a biologically plausible recurrent spiking neural network where the output population recodes its inputs to produce spikes that are independently decodeable. We show that this network can be learned in a supervised manner by a simple local learning rule.

Pavlovian predictions of future aversive outcomes lead to behavioral inhibition, suppression, and withdrawal. There is considerable evidence for the involvement of serotonin in both the learning of these predictions and the inhibitory consequences that ensue, although less for a causal relationship between the two. In the context of a highly simplified model of chains of affectively charged thoughts, we interpret the combined effects of serotonin in terms of pruning a tree of possible decisions, (i.e., eliminating those choices that have low or negative expected outcomes). We show how a drop in behavioral inhibition, putatively resulting from an experimentally or psychiatrically influenced drop in serotonin, could result in unexpectedly large negative prediction errors and a significant aversive shift in reinforcement statistics. We suggest an interpretation of this finding that helps dissolve the apparent contradiction between the fact that inhibition of serotonin reuptake is the first-line treatment of depression, although serotonin itself is most strongly linked with aversive rather than appetitive outcomes and predictions.

Uncertainty coming from the noise in its neurons and the ill-posed nature of many tasks plagues neural computations. Maybe surprisingly, many studies show that the brain manipulates these forms of uncertainty in a probabilistically consistent and normative manner, and there is now a rich theoretical literature on the capabilities of populations of neurons to implement computations in the face of uncertainty. However, one major facet of uncertainty has received comparatively little attention: time. In a dynamic, rapidly changing world, data are only temporarily relevant. Here, we analyze the computational consequences of encoding stimulus trajectories in populations of neurons. For the most obvious, simple, instantaneous encoder, the correlations induced by natural, smooth stimuli engender a decoder that requires access to information that is nonlocal both in time and across neurons. This formally amounts to a ruinous representation. We show that there is an alternative encoder that is computationally and representationally powerful in which each spike contributes independent information; it is independently decodable, in other words. We suggest this as an appropriate foundation for understanding time-varying population codes. Furthermore, we show how adaptation to temporal stimulus statistics emerges directly from the demands of simple decoding.

Our understanding of the input-output function of single cells has been substantially advanced by biophysically accurate multi-compartmental models. The large number of parameters needing hand tuning in these models has, however, somewhat hampered their applicability and interpretability. Here we propose a simple and well-founded method for automatic estimation of many of these key parameters: 1) the spatial distribution of channel densities on the cell's membrane; 2) the spatiotemporal pattern of synaptic input; 3) the channels' reversal potentials; 4) the intercompartmental conductances; and 5) the noise level in each compartment. We assume experimental access to: a) the spatiotemporal voltage signal in the dendrite (or some contiguous subpart thereof, e.g. via voltage sensitive imaging techniques), b) an approximate kinetic description of the channels and synapses present in each compartment, and c) the morphology of the part of the neuron under investigation. The key observation is that, given data a)-c), all of the parameters 1)-4) may be simultaneously inferred by a version of constrained linear regression; this regression, in turn, is efficiently solved using standard algorithms, without any "local minima" problems despite the large number of parameters and complex dynamics. The noise level 5) may also be estimated by standard techniques. We demonstrate the method's accuracy on several model datasets, and describe techniques for quantifying the uncertainty in our estimates.

As animals interact with their environments, they must constantly update estimates about their states. Bayesian models combine prior probabil- ities, a dynamical model and sensory evidence to update estimates op- timally. These models are consistent with the results of many diverse psychophysical studies. However, little is known about the neural rep- resentation and manipulation of such Bayesian information, particularly in populations of spiking neurons. We consider this issue, suggesting a model based on standard neural architecture and activations. We illus- trate the approach on a simple random walk example, and apply it to a sensorimotor integration task that provides a particularly compelling example of dynamic probabilistic computation. Bayesian models have been used to explain a gamut of experimental results in tasks which require estimates to be derived from multiple sensory cues. These include a wide range of psychophysical studies of perception;13 motor action;7 and decision-making.3, 5 Central to Bayesian inference is that computations are sensitive to uncertainties about afferent and efferent quantities, arising from ignorance, noise, or inherent ambiguity (e.g., the aperture problem), and that these uncertainties change over time as information accumulates and dissipates. Understanding how neurons represent and manipulate uncertain quantities is therefore key to understanding the neural instantiation of these Bayesian inferences. Most previous work on representing probabilistic inference in neural populations has fo- cused on the representation of static information.1, 12, 15 These encompass various strategies for encoding and decoding uncertain quantities, but do not readily generalize to real-world dynamic information processing tasks, particularly the most interesting cases with stim- uli changing over the same timescale as spiking itself.11 Notable exceptions are the re- cent, seminal, but, as we argue, representationally restricted, models proposed by Gold and Shadlen,5 Rao,10 and Deneve.4 In this paper, we first show how probabilistic information varying over time can be repre- sented in a spiking population code. Second, we present a method for producing spiking codes that facilitate further processing of the probabilistic information. Finally, we show the utility of this method by applying it to a temporal sensorimotor integration task. 1 TRAJECTORY ENCODING AND DECODING We assume that population spikes R(t) arise stochastically in relation to the trajectory X(t) of an underlying (but hidden) variable. We use RT and XT for the whole trajectory and spike trains respectively from time 0 to T . The spikes RT constitute the observations and are assumed to be probabilistically related to the signal by a tuning function f (X, i): P (R(i, T )|X(T )) f (X, i) (1) for the spike train of the ith neuron, with parameters i. Therefore, via standard Bayesian inference, RT determines a distribution over the hidden variable at time T , P (X(T )|RT ). We first consider a version of the dynamics and input coding that permits an analytical examination of the impact of spikes. Let X(t) follow a stationary Gaussian process such that the joint distribution P (X(t1), X(t2), . . . , X(tm)) is Gaussian for any finite collection of times, with a covariance matrix which depends on time differences: Ctt = c(|t - t |). Function c(|t|) controls the smoothness of the resulting random walks. Then, P (X(T )|RT ) p(X(T )) dX(T )P (R X(T ) T |X(T ))P (X(T )|X (T )) (2) where P (X(T )|X(T )) is the distribution over the whole trajectory X(T ) conditional on the value of X(T ) at its end point. If RT are a set of conditionally independent inhomoge- neous Poisson processes, we have P (RT |X(T )) f (X(t d f (X( ), i i ), i) exp - i i) , (3) where ti are the spike times of neuron i in RT . Let = [X(ti )] be the vector of stimulus positions at the times at which we observed a spike and = [(ti )] be the vector of spike positions. If the tuning functions are Gaussian f (X, i) exp(-(X - i)2/22) and sufficiently dense that d f (X, i i) is independent of X (a standard assumption in population coding), then P (RT |X(T )) exp(- - 2/22) and in Equation 2, we can marginalize out X(T ) except at the spike times ti : P (X(T )|RT ) p(X(T )) d exp -[, X(T )]T C-1 [, X(T )] - - 2 (4) 2 22 and C is the block covariance matrix between X(ti ), x(T ) at the spike times [tt ] and the final time T . This Gaussian integral has P (X(T )|RT ) N ((T ), (T )), with (T ) = CT t(Ctt + I2)-1 = k (T ) = CT T - kCtT (5) CT T is the T, T th element of the covariance matrix and CT t is similarly a row vector. The dependence in on past spike times is specified chiefly by the inverse covariance matrix, and acts as an effective kernel (k). This kernel is not stationary, since it depends on factors such as the local density of spiking in the spike train RT . For example, consider where X(t) evolves according to a diffusion process with drift: dX = -Xdt + dN (t) (6) where prevents it from wandering too far, N (t) is white Gaussian noise with mean zero and 2 variance. Figure 1A shows sample kernels for this process. Inspection of Figure 1A reveals some important traits. First, the monotonically decreasing kernel magnitude as the time span between the spike and the current time T grows matches the intuition that recent spikes play a more significant role in determining the posterior over X(T ). Second, the kernel is nearly exponential, with a time constant that depends on the time constant of the covariance function and the density of the spikes; two settings of these parameters produced the two groupings of kernels in the figure. Finally, the fully adaptive kernel k can be locally well approximated by a metronomic kernel k (shown in red in Figure 1A) that assumes regular spiking. This takes advantage of the general fact, indicated by the grouping of kernels, that the kernel depends weakly on the actual spike pattern, but strongly on the average rate. The merits of the metronomic kernel are that it is stationary and only depends on a single mean rate rather than the full spike train RT . It also justifies Kernels k and ks Variance ratio Full kernel A B D -0.5 -2 10 10 2 / 0 -4 2 5 Space 10 Kernel size (weight) 0 0.5 0 0.03 0.06 0.09 0.04 0.06 0.08 0.1 t-tspike Time C True stimulus and means Regular, stationary kernel E 0.5 -0.5 0 0 Space Space -0.5 0.5 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time Time Figure 1: Exact and approximate spike decoding with the Gaussian process prior. Spikes are shown in yellow, the true stimulus in green, and P (X(T )|RT ) in gray. Blue: exact inference with nonstationary and red: approximate inference with regular spiking. A Ker- nel samples for a diffusion process as defined by equations 5, 6. B, C: Mean and variance of the inference. D: Exact inference with full kernel k and E: approximation based on metronomic kernel k. (Equation 7). the form of decoder used for the network model in the next section.6 Figure 1D shows an example of how well Equation 5 specifies a distribution over X(t) through very few spikes. Finally, 1E shows a factorized approximation with the stationary kernel similar to that used by Hinton and Brown6 and in our recurrent network: ^ t P (X(t)|R(t)) f (X, kst j=0 j ij = exp(-E(X(t), R(t), t)), (7) i i) By design, the mean is captured very well, but not the variance, which in this example grows too rapidly for long interspike intervals (Figure 1B, C). Using a slower kernel im- proves performance on the variance, but at the expense of the mean. We thus turn to the net- work model with recurrent connections that are available to reinstate the spike-conditional characteristics of the full kernel. 2 NETWORK MODEL FORMULATION Above we considered how population spikes RT specify a distribution over X(T ). We now extend this to consider how interconnected populations of neurons can specify distributions over time-varying variables. We frame the problem and our approach in terms of a two-level network, connecting one population of neurons to another; this construction is intended to apply to any level of processing. The network maps input population spikes R(t) to output population spikes S(t), where input and output evolve over time. As with the input spikes, ST indicates the output spike trains from time 0 to T , and these output spikes are assumed to determine a distribution over a related hidden variable. For the recurrent and feedforward computation in the network, we start with the de- ceptively simple goal9 of producing output spikes in such a way that the distribution Q(X(T )|ST ) they imply over the same hidden variable X(T ) as the input, faithfully matches P (X(T )|RT ). This might seem a strange goal, since one could surely just lis- ten to the input spikes. However, in order for the output spikes to track the hidden variable, the dynamics of the interactions between the neurons must explicitly capture the dynamics of the process X(T ). Once this `identity mapping' problem has been solved, more general, complex computations can be performed with ease. We illustrate this on a multisensory integration task, tracking a hidden variable that depends on multiple sensory cues. The aim of the recurrent network is to take the spikes R(t) as inputs, and produce output spikes that capture the probabilistic dynamics. We proceed in two steps. We first consider the probabilistic decoding process which turns ST into Q(X(t)|ST ). Then we discuss the recurrent and feedforward processing that produce appropriate ST given this decoder. Note that this decoding process is not required for the network processing; it instead provides a computational objective for the spiking dynamics in the system. We use a simple log-linear decoder based on a spatiotemporal kernel:6 Q(X(T )|ST ) exp(-E(X(T ), ST , T )) , where (8) E(X, S T T , T ) = S(j, T - ) j =0 j (X, ) (9) is an energy function, and the spatiotemporal kernels are assumed separable: j(X, ) = gj(X)( ). The spatial kernel gj(X) is related to the receptive field f (X, j) of neuron j and the temporal kernel j(X, ) to k The dynamics of processing in the network follows a standard recurrent neural architecture for modeling cortical responses, in the case that network inputs R(t) and outputs S(t) are spikes. The effect of a spike on other neurons in the network is assumed to have some simple temporal dynamics, described here again by the temporal kernel ( ): ri(t) = T R(i, T - )( ) s S(j, T - )( ) =0 j (t) = T =0 where T is the extent of the kernel. The response of an output neuron is governed by a stochastic spiking rule, where the probability that neuron j spikes at time t is given by: P (S(j, t) = 1) = (uj(t)) = ( w v i ij ri(t) + k kj sk (t - 1)) (10) where () is the logistic function, and W and V are the feedforward and recurrent weights. If ( ) = exp(- ), then uj(T ) = (0)(Wj R(T ) + Vj S(T )) + (1)uj(T - 1); this corresponds to a discretization of the standard dynamics for the membrane potential of a leaky integrate-and-fire neuron: duj = -u dt j +WR+VS, where the leak is determined by the temporal kernel. The task of the network is to make Q(X(T )|ST ) of Equation 8 match P (X(T )|RT ) com- ing from one of the two models above (exact dynamic or approximate stationary kernel). We measure the discrepancy using the Kullback-Leibler (KL) divergence: J = KL [P (X(T )|R t T )||Q(X (T )|ST )] (11) and, as a proof of principle in the experiments below, find optimal W and V by minimizing the KL divergence J using back-propagation through time (BPTT). In or- der to implement this in the most straightforward way, we convert the stochastic spik- ing rule (Equation 10) to a deterministic rule via the mean-field assumption: Sj(t) = ( w v i ij ri(t) + k kj sk (t - 1)). The gradients are tedious, but can be neatly ex- pressed in a temporally recursive form. Note that our current focus in the system is on the representational capability of the system, rather than its learning. Our results establish that the system can faithfully represent the posterior distribution. We return to the issue of more plausible learning rules below. The resulting network can be seen as a dynamic spiking analogue of the recurrent network scheme of Pouget et al.:9 both methods formulate feedforward and recurrent connections so that a simple decoding of the output can match optimal but complex decoding applied to the inputs. A further advantage of the scheme proposed here is that it facilitates downstream processing of the probabilistic information, as the objective encourages the formation of distributions at the output that factorize across the units. 3 RELATED MODELS Ideas about the representation of probabilistic information in spiking neurons are in vogue. One treatment considers Poisson spiking in populations with regular tuning functions, as- suming that stimuli change slowly compared with the inter-spike intervals.8 This leads to a Kalman filter account with much formal similarity to the models of P (X(T )|RT ). However, because of the slow timescale, recurrent dynamics can be allowed to settle to an underlying attractor. In another approach, the spiking activity of either a single neuron4 or a pair of neurons5 is considered as reporting (logarithmic) probabilistic information about an underlying binary hypothesis. A third treatment proposes that a population of neurons directly represents the (logarithmic) probability over the state of a hidden Markov model.10 Our method is closely related to the latter two models. Like Deneve's4 we consider the transformation of input spikes to output spikes with a fixed assumed decoding scheme so that the dynamics of an underlying process is captured. Our decoding mechanism produces something like the predictive coding apparent in Deneve's scheme, except that here, a neu- ron may not need to spike not only if it itself has recently spiked and thereby conveyed the appropriate information; but also if one of its population neighbors has recently spiked. This is explicitly captured by the recurrent interactions among the population. Our scheme also resembles Rao's10 approach in that it involves population codes. Our representational scheme is more general, however, in that the spatiotemporal decoder defines the relation- ship between output spikes and Q(X(T )|ST ), whereas his method assumes a direct en- coding, with each output neuron's activity proportional to log Q(X(T )|ST ). 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