Recently, there have been remarkable advances in modeling the relationships between the sensory environment, neuronal responses, and behavior. However, most models cannot encompass variable stimulus-response relationships such as varying response latencies and state or context dependence of the neural code. Here, we consider response modeling as a dynamic alignment problem and model stimulus and response jointly by a mixed pair hidden Markov model (MPH). In MPHs, multiple stimulus-response relationships (e. g. , receptive fields) are represented by different states or groups of states in a Markov chain. Each stimulus-response relationship features temporal flexibility, allowing modeling of variable response latencies, including noisy ones. We derive algorithms for learning of MPH parameters and for inference of spike response probabilities. We show that some linear-nonlinear Poisson cascade (LNP) models are a special case of MPHs. We demonstrate the efficiency and usefulness of MPHs in simulations of both jittered and switching spike responses to white noise and natural stimuli. Furthermore, we apply MPHs to extracellular single and multi-unit data recorded in cortical brain areas of singing birds to showcase a novel method for estimating response lag distributions. MPHs allow simultaneous estimation of receptive fields, latency statistics, and hidden state dynamics and so can help to uncover complex stimulus response relationships that are subject to variable timing and involve diverse neural codes.
Neural response models are used to relate neural activity to sensory stimuli and motor behavior. A very common type of neural response model is comprised of a linear stage, at which one or more linear filters (often referred to as receptive fields) are applied to the stimulus, and a subsequent non-linear stage that converts the filter outputs into a spiking probability that feeds into a Poisson process generating the spikes [1]. More precisely, the spiking probability (i. e. , the instantaneous rate of the Poisson process) of a neuron is modeled as, where the column vector represents the stimulus, the nonlinearity, and is a row vector containing the linear filter or a matrix in case of several filters. Variations of these linear-nonlinear Poisson cascade models (LNP models) have been studied extensively [2]–[5]. Parameter estimation techniques range from spike triggered averaging in case of one linear filter and white noise stimuli, to spike triggered covariance [1], [6] in case of several linear filters, and maximally informative dimensions, in case of one or several linear filters and no restrictions on the distribution of stimuli [2], [4]. Although these techniques are effective in many domains, they fail in others, where the neural code might be more intricate (detailed below). A crucial assumption about the relationship between stimulus and response inherent in these techniques is that the response latency of the cell, the filters (or receptive fields), and the non-linearity all are assumed to be the same throughout the experiment (Fig. 1A). This assumption of a fixed stimulus-response relationship is, however, not necessarily valid. On the one hand, the relationship between stimulus and response, the neural code, could vary in time (Fig. 1C). On the other hand, the response latency could be noisy or vary systematically (Fig. 1A). The extent to which a fixed stimulus-response relationship applies to neurophysiological data is unclear. In terms of spike timing, a fixed relationship entails both constant response latency and an amount of spike-time-jitter that is smaller than the relevant temporal structure of the receptive field. However, noisy response latencies (Fig. 1A) are observed in almost all electrophysiological studies because neural systems are intrinsically noisy. Variations in response latency to a repeated stimulus (measured as the standard deviation of time of first spike after stimulus onset) in the range of 3–5 ms have been reported already at a very low stage of the visual system, in retinal ganglion cells [7]. Variability in response latency can be notably larger in cortical areas. For instance, variations in first-spike latency (again measured as the standard deviation) of up to 12. 5 ms have been observed in single cells in ferret primary visual cortex in response to flashed natural images [8]. Furthermore, systematically varying response latencies have been demonstrated in various model systems, e. g. , image contrast modulates response latency both in retinal ganglion cells [9] and in visual cortical neurons [10]–[12], fueling discussions about the role of spike latency in neural coding [12], [13]. In a recent study, latencies of cells in macaque inferotemporal cortex were found to systematically differ for primate and non-primate face stimuli, with latency differences on the order of tens of milliseconds [14]. When latencies are strongly fluctuating or spike time-jitter is large, many modeling techniques that assume a fixed stimulus-response relationship, such as spike triggered averaging, yield suboptimal results [15], [16]: The estimated receptive fields are blurred and the accuracy of predicted responses to novel stimuli is low (Fig. 1B). Fixed stimulus-response relationships can also be violated in case of changes in intrinsic or hidden brain states [17], [18]. For example, neurons in the primary somatosensory cortex of the rat undergo up and down states given by two separate membrane potentials. Spiking responses to whisker deflections in these cells are dependent on whether neurons are in the up or the down state: in the down state, a reliable response is observed, whereas in the up state activity is largely stimulus independent [19]. Stimulus context can also induce changes in internal states. For instance, in an awake marmoset study of single-unit responses in the auditory cortex to sequences of 2 sound stimuli, responses to the second stimulus were not static but depended strongly on the first stimulus. This modulation in second-stimulus responses can last longer than 1. 5 seconds [20]. We illustrate dependence of neural computation on intrinsic states in a cartoon (Fig. 1C) in which the simulated neuron switches between 2 static receptive fields. Response switching has also been observed in the amphibian retina. Ganglion-cell activity is typically dominated by OFF responses. However, a large peripheral image shift (as occurs during head saccades) can induce a switch (for a few hundred milliseconds) from transmitting OFF signals to transmitting ON signals [21]. One of the most compelling examples of response switching has been observed in songbirds: many neurons in cortical motor and auditory areas are responsive to playback of sound stimuli except when birds are singing, at which times responses are locked to the song but not influenced by sound playback [22], [23]. Hence, if such neural responses were to be modeled across singing and non-singing states, anything but a two-state model would be inadequate. Indeed, many classical models fail in cases of response switching: the estimated classical receptive fields contain superimposed structures derived from the switched responses, which yields sub-optimal results (Fig. 1D). To address variable response latencies and dynamic neural codes we consider the problem of neural response modeling as an alignment problem. We introduce mixed pair hidden Markov models (MPHs) as novel neural response models allowing for dynamic alignment of stimulus and response without fixed stimulus-response assumptions. In the case of varying spike latencies (e. g. when a neuron fires in response to a particular stimulus but with a variable latency or lag, Fig. 1A, B), MPHs help to detect corresponding stimulus-response parts by associating individual spikes with particular stimulus time points; and, they help to uncover stimulus-response relationships including the spike-jitter statistics and the receptive field of the neuron. In case of switching dynamics (e. g. , when a neuron switches between being responsive to either one stimulus or another depending on the behavioral state of the animal or a cueing stimulus, Fig. 1C), MPHs help to identify parameters such as the receptive-field switching probabilities and the switching events. Our MPH approach to dynamic alignment combines response switching (context dependency) and spike-time jitter or systematically varying latencies (flexible timing) in one unified framework. We show how to use stimuli and neural responses to jointly estimate all model parameters including spike time jitter, systematically varying latencies, and switching probabilities. We demonstrate the benefits of dynamic alignment on simulated data and on extracellular data recorded in cortical brain areas of singing birds.
We solve the alignment problem by jointly modeling stimulus and response by a mixed pair hidden Markov model (MPH), which is a generative model for both the stimulus and neural response. In MPHs, different neural codes, i. e. different relationships between neural activity and sensory input, can coexist as different states or groups of states in a Markov chain. MPHs are unlike classical hidden Markov models because they dynamically operate on pairs of sequences - neural activity and stimulus - instead of single sequences (i. e. a joint sequence of neural activity and stimulus). For a mathematically detailed introduction to MPHs (and an introduction to HMMs), see the Materials and Methods section. We explain the workings of MPHs in intuitive terms by considering first the special case of jittered spike times (Fig. 1A). We assume spikes are associated with the stimulus (i. e. a time window of the stimulus [1]) that precede the spikes by an average time lag. Instead of associating spikes and stimuli at a constant lag (such as is the case for standard spike triggered methods including STA, STC, maximally informative dimensions, etc. .), MPHs associate an individual spike with a stimulus at the individual lag (in units of stimulus-response bins, being an integer, can be different for each spike). MPHs achieve this flexibility via three different types of hidden states and by keeping track of the momentary lag and its evolution. First, matching states (M-states) associate a spike with a stimulus at the current lag by modeling the joint probability distribution of spike and stimulus (Fig. 2A, middle). The simplest case are Gaussian stimulus models comprising two Gaussians, one of which models stimuli jointly occurring with spikes and the other models stimuli not occurring with spikes (Fig. 2A, middle). A model with only a single such M-state is bound to a fixed lag and is equivalent to an LNP model (under appropriate parameter constraints, see also the section on LNP equivalence below). To achieve a variable lag, we introduce two more types of states: X-states (X stands for the stimulus sequence) and R-states (R stands for response sequence). These states can change the momentary lag as follows. An X-state models the stimulus (but not the response) via some probability distribution, for instance a Gaussian (Fig. 2A, left). The X-state changes the current lag from to. Analogously, an R-state models only the spiking response (but not the stimulus) via a discrete probability distribution (e. g. spike or no-spike, Fig. 2A, right). An R-state changes the current lag from to. An MPH consisting of an M, X, and R state can thus model a spike-stimulus pair via the M state or via the X and the R state. In general fewer states are preferred as there is a cost associated with switching from one state to another. Thus, an MPH will try to keep the lag between stimulus and spike constant unless there is evidence for changing the lag via X and R states. Intuitively, one can think of an MPH as a finite state automaton that processes symbols from two sequences at the same time, the stimulus (X-sequence) and the response sequence (R-sequence). Under this analogy, M-states process one symbol of each sequence (they match a symbol pair from X and R), X-states process only a stimulus symbol, and R-states process only a response-symbol. The automaton keeps track of two pointers that indicate the current position in the stimulus sequence as well as the current position in the response sequence. The pointer difference corresponds to the current lag dT+e (the “automaton” considers all possible lags, weighted probabilistically). A sequence of hidden states in an MXR-MPH (M, X, and R states) can be depicted as a path in an alignment matrix that spans all possible pairings between stimulus and response (Fig. 2B): An M-State corresponds to diagonal movement along the matrix from position, i. e. position in the stimulus sequence and in the response sequence, to position. An X-State corresponds to horizontal movement from position to and an R-State corresponds to vertical movement from position to. A change in the temporal relationship between stimulus and response (i. e. spike jitter) is reflected in non-diagonal (horizontal or vertical) steps in the alignment matrix (with step size provided by the discretization of stimulus and neural response sequences, Fig. 2B). This 3-state model will be applied to simulated and real data in the next section. In order to handle state-dependent (switching) neural responses (Fig. 1C), we consider MPHs with several M-states (Fig. 2C). Multiple M states add another dimension to the alignment matrix, which we henceforth call alignment tensor (Fig. 2D). The simplest switching-enabling MPH has two M-states but no X or R states. Each M-state is associated with a particular receptive field to be estimated (here we use ‘receptive field’ in the most general way, independent of linearity and related assumptions). With several M states, spikes can be associated to stimuli via one of several joint probability distributions over spike and stimulus. Probabilistic transitions between M states allow the MPH to switch between receptive fields, which is shown on simulated and real data below. The general MPH has several M-, X-, and R-States and thus simultaneously permits flexible timing and context dependency. The parameters defining probabilistic transitions into and out of hidden states are: The parameters defining emission probabilities are: As emission probability densities associated with X and M states we use multivariate Gaussians or mixtures of Gaussians, respectively. For hidden X-state we write the emission probability density aswhere is the weight of the kth mixture component, denotes the total number of mixture components (which may vary for different hidden states, i. e. some can be zero), and and denote the Gaussian mean and covariance matrix of the kth mixture component. For M states we keep track of one such multivariate Gaussian for each response state (each response state is associated with a distinct stimulus emission). In the following we present detailed analyses of the spike-jitter and response-switching MPH architectures. First, we discuss an MPH with only one M-state (M-MPH) and multivariate Gaussian stimulus models. We show that, under appropriate parameter constraints, such a model is equivalent to a 1-dimensional LNP model and describe its relation to linear regression and linear discriminant analysis and the resulting strengths and limitations. Second, we discuss an extension of that M-MPH to a model that also possesses an X- and an R-state (MXR-MPH). We illustrate in simulations how this model can account for spike time jitter and varying latencies on white noise and on natural stimuli. Third, we treat an extension of the M-MPH to multiple M-states (Mn-MPH) and illustrate through simulations how this model can account for switching dynamics and context dependency. All of these models can be cascaded with an additional non-linearity so that they form NNP cascades (as opposed to LNP cascades). In the chapter that then follows, we apply these models to data recorded from single units of cells in the behaving bird. In the following we demonstrate the ability of MXR-MPHs (Fig. 2A) to recover the correct receptive field (RF) on simulated data with noisy latencies, i. e. spike-time-jitter. When predicting spiking probabilities on novel data, the MXR-MPH outperforms purely spike-triggered methods. For the MXR-MPH, we denote the stimulus means and covariance matrices in the M-State by (non-spiking) and (spiking) and in the in the X-State by. In accordance with the section on LNP equivalence and to simplify this general MPH, we introduce the following parameter constraints: First, we fix all covariance matrices to identity matrices: Second, we fix the means and to zero (equal to the mean of the stimulus ensemble). Third, we do not allow the R-state to generate spikes (zero spike emission probability) because we require that each spike is matched to a stimulus (note that if we allowed the R-state to generate spikes, the model would distinguish between spikes generated by the M-state and spikes generated by the R-state — such distinction could be used for distinguishing stimulus driven from spontaneous activity, which was not our focus). Given these parameter constraints, the remaining free parameters in the model are the receptive field (the mean of the spike triggered stimulus ensemble, Fig. 2C), and the transition probabilities among X, R, and M-state. After training the MPH, the jitter statistics are implicit in the model' s parameters; below we show how to explicitly compute the jitter statistics for natural stimuli. MPHs with several M states support context dependency. They can model multiple stimulus-response relationships associated, for instance, with distinct behavioral states of an animal. To demonstrate this flexibility of MPHs, we simulated a neuron that randomly switches (according to a Markov process with equal probabilities) between two linear-nonlinear models (each defined as in the previous section), i. e. , neural responses were governed by a hidden state sequence that determined which receptive field was active at any given time (Fig. 5A). We generated responses of this artificial neuron to 100 white noise sequences, each spanning 1000 time bins (arbitrary timescale), and 21 dimensions or stimulus channels. For each sequence, spike responses were generated using the switching LNP model, resulting in a total of 4374 spikes on average (mean 0. 044 spikes/bin). We generated and tested data for different RF combinations, characterized by different rotations in the plane of one of the RFs (Fig. 5B, left). To uncover the hidden switching dynamics and the two RFs, we trained a Gaussian M2-MPH with two M-states and (Fig. 2C). Gaussian parameters were constrained in the following way: First, we imposed zero means: ; and second, we fixed all covariance matrices to the identity matrix. The remaining free parameters of the model were the means and of the spike-triggered ensembles (i. e. , the receptive fields) and the transition probabilities between the two states (which reflect the switching statistics). We compared the MPH with STA and STC [1], [6] models. For all three models, we computed RF estimates and the response prediction performance. The trained MPH faithfully recovered both RFs, whereas the (single) RF estimated of STA consisted of a superposition of the two RFs, and the RFs estimated using STC were severely corrupted by noise in most cases (Fig. 5B). As a result, the MPH predicted responses better on an independent validation set than did STA and STC (Fig. 5C, averaged over 10 runs). We assessed the quality of the recovered RFs of all three methods and all 37 tested rotations by matching each original RF to the recovered RF with smallest distance and by averaging the two distances. The MPH recovers the RF much better then STA or STC (Fig. 5D, average over 10 runs; drops in MPH-reconstruction quality are due to local minima, compare figure text). The degraded performance of the STC model has two reasons. First, the covariance of the spike triggered ensemble needs to be reliably estimated (with quadratically many degrees of freedom as there are stimulus dimensions compared to a linear number of degrees of freedom for the M2-MPH). Second, the linear filters uncovered by STC are orthogonal [1], whereas the M2-MPH is not constrained in this way. It is possible to show that the M2-MPH firing rate to a stimulus is given bywhere are two non-linearities and and are the prior probabilities of hidden states and, respectively. The STA model, on the other hand, is bound to model firing rates as An extreme example that illustrates the failure of RF estimation with STAs is a neural response model that pools over two filters and. In that case the estimated RF using STA is uniform and has no predictive power at all, unlike the MPH (e. g. Fig. 5C, rotation angle 180°). A less extreme but potentially more relevant case is that of complex cells in primary visual cortex with overlapping excitatory and inhibitory oriented receptive subfields (such cells are often modeled by pooling over four oriented filters that are phase shifted, , and, respectively [29]–[31]). A switching M4-MPH with four M-states can recover these phase shifted filters, whereas STA yields only a blurred RF. To demonstrate that MPHs work well in practice even when the amount of available data is small and the true spike generating process is unknown, we apply the MXR- and the Mn-MPH to extracellular spike data recorded in the forebrain nucleus interface of the nidopallium (NIf) of songbirds (Fig. 6A). NIf is a higher-order song-control nucleus; lesion and inactivation studies have shown that NIf exhibits both sensory and motor functions [32]–[35]. Multi-unit NIf activity is generally strongest shortly before and during syllable production and weakest during the times corresponding to silent intervals between syllables [36], [37]. These findings suggest a pre-vocal role of NIf spikes during song, prompting us to expect in singing birds a negative latency of NIf spikes relative to song (spikes precede sounds, as opposed to a positive latency that would result if NIf firing was sensory during vocal production). Due to the difficulty of recording in singing animals, available spike trains are relatively short (the average total spike train duration was 73 s per cell) and contain few spikes (∼1500 spikes per cell). To investigate latencies of NIf single-unit spikes relative to song, we first fitted an LNP model using reverse correlation (RC, Fig. 6C, left). To overcome problems of over-fitting (due to the limited amount of data available) we used a regularized version of the stimulus covariance matrix: (7) where denotes the number of stimulus dimensions, denotes the unregularized stimulus covariance matrix and denotes its normalized trace (such regularization yielded better generalization performance). Next, we trained a MXR-MPH on large 0. 25 s song spectrogram windows (with covariance matrices in M and X states fixed to the regularized stimulus covariance matrix in Eq. 7). The MPH RF was similar to the reverse correlation RF (Fig. 6B), but it reflected more clearly that the cell fired before sounds and not thereafter (consider for example the stronger inhibitory band near 10 ms). MPH and reverse correlation encoding performances on a test set were comparable (Fig. 6D, left data points). Note that by construction, differences between MXR-MPH and reverse correlation RFs arise from spike-time-jitter. To characterize response latencies (and jitter) we estimated probability distributions of the temporal offset between stimulus and response in M-states via the alignment kernel (Fig. 6B, left). Negative lags in the alignment kernel imply that spikes occur before corresponding events in the stimulus, whereas positive lags imply that spikes occur thereafter. The alignment kernel was centered at a small negative time lag and exhibited a small temporal spread, revealing high temporal precision of NIf spike trains. Predicted responses (5-fold cross validation) for the reverse correlation model and the MXR-MPH were equally good (Fig. 6D, left data points), confirming high temporal precision of NIf spike trains. The MPH allowed us to strongly reduce model complexity by shrinking linear filters (RF sizes) down to less than 30 ms. For such short RFs, the cell latency is reflected entirely in the alignment kernel. Based on the RF estimate in Fig. 6B and 6C, we expected the jitter kernel to be centered near −30 to −40 ms. Indeed, the kernel peaked near −40 ms (Fig. 6C, right), implying that the MPH aligned spikes to portions of the stimulus occuring about 40 ms after the spike, suggesting a premotor function of this cell and thus agreeing with the hypothesized premotor function of NIf. Additionally we trained an Mn-MPH with various numbers of states on the same NIf cell (unlike for the M2-MPH applied in the section on switching dynamics we did not constrain the means). The Mn-MPH showed modest improvements over reverse correlation, its peak validation CC occurred at 8 states (Fig. 6D), suggesting that this NIf cell fires prior to several distinct song features. We also analyzed data for another recording site in NIf, composed of 54 s of singing with concurrent spiking (1659 spikes, about 60 stereotyped song motifs). The RF estimated using reverse correlation (Fig 6E, left) revealed diffuse spectrotemporal tuning, making it difficult to decide whether this cell is sensory or motor in function. By contrast, the MPH alignment kernel (Fig. 6E, right) quite clearly revealed a motor function in this cell, evidenced by the predominance of negative alignment shifts. The MPH RF showed a rather narrow frequency tuning near 2,6 kHz (Fig. 6E, middle). Encoding performance for the MXR-MPH with large RF was again similar to reverse correlation (Fig. 6H, left data points), yet an Mn-MPH yielded slightly superior performance (Fig. 6H, right data points).
We introduced a novel technique for neural response modeling and receptive field estimation that overcomes limitations of fixed stimulus-response relationships. We proposed to view neural coding as an alignment problem that can be tackled by mixed pair hidden Markov models (MPHs), which jointly model stimulus and response and can naturally account for noisy or systematically varying latencies as well as for context dependent neural codes that depend on internal (hidden) states. Discrete pair HMMs have been used in the context of gene alignment to find corresponding parts in related gene sequences [38]. To our knowledge they have not yet been applied to response modeling. We demonstrated that simple MPHs with Gaussian stimulus models and a fixed shared covariance matrix are equivalent to one dimensional LNP models with sigmoid non-linearity and we extended these basic MPHs to allow flexible timing and context dependency. Thereby MPHs endow standard RF estimation techniques such as spike triggered averaging (STA) and reverse correlation with flexible timing and context dependency. We tested our approach on simulated and real data and demonstrated the benefits of alignment in terms of improved predictability of simulated and real neural responses, improved receptive field estimates as well as the capability of estimating jitter latency statistics and switching states. Key properties of MPHs are: 1) X- and R-states that model stimulus or response alone and allow for flexible timing via dynamic temporal alignment, and 2) M-states that allow for context dependency via model switching. Using our estimation techniques, these three types of states can be freely combined in a highly flexible approach to neural coding and decoding without the need to develop additional algorithms. We derived MPH parameters estimation updates for Gaussian mixture models with unrestricted covariance matrices (Materials and Methods). The (non-mixture) Gaussian MPHs we studied performed well in simulations (including natural stimuli), even though the assumption of Gaussian stimulus models can be violated by natural stimuli [24], [39]. In these cases, mixtures of Gaussians can be useful to approximate arbitrary stimulus distributions and overcome problems of receptive field biases [27]. Beyond mixtures of Gaussians, EM update equations for other mixture families are known as well [40] and could be adapted to MPHs. Other modeling approaches have been pursued to estimate neural responses in the presence of spike time jitter [15], [16], [41]. One approach is to simultaneously estimate the jitter distribution and the RF using the EM algorithm [16]. This technique has been successfully applied to white noise stimuli (identity covariance) [16], but not to stimuli with non-identity covariance, i. e. natural stimuli. Furthermore, in [16] the jitter of adjacent spikes is assumed to be independent – an assumption that might be violated in cases where jitter depends on slowly varying internal states or is correlated for other reasons. The dynamic alignment technique we present here generalizes these approaches in two ways. First, in MPHs there is no need to constrain the stimulus covariance matrix, so that natural stimuli can be readily processed. Second, MPHs can account for correlated as well as uncorrelated jitter among adjacent or nearby spikes and thus allow modeling of both systematically and slowly varying spike latencies. Furthermore, in [16] the jitter distribution is explicitly assumed to be of Gaussian form whereas the jitter distribution of the MPH is implicit in the transition probabilities and has degrees of freedom commensurate with the number of hidden states and their transitions. The ability of MPHs to emulate switching models is particularly useful given that switching dynamics are important in many neural systems. A number of other approaches have been introduced to handle response switching and context dependency. Several of them are based on hidden Markov models [42]–[48]. The hidden states in these models typically reflect neural activity but not the stimulus. Models with hidden states that reflect both stimulus and response, such as switching Kalman filters [49] or generalized linear models with hidden states [50], [51], have also been proposed. These models are similar to MPHs with only M-states but no X- and R-states. Furthermore, our approach extends these models in that stimulus-response relationships within each hidden state can be quadratic (single Gaussians, unconstrained covariance matrices) or formed by Gaussian mixtures. Another way of modeling context dependencies are “multi-linear” models encompassing a multiplicative context term (by itself modeled through a “multi-linear” model) that depends on the projection of the stimulus (in some time window) onto a set of basis functions [52]. MPHs complement such approaches by allowing more complex types of contextual influence via the underlying Markov structure. This is also an advantage over techniques like spike triggered covariance that can recover multiple filters [1], [6], [53]–[55] but cannot attribute Markovian dynamics to the individual filters. For instance, MPHs allow for context effects over very long time scales, context effects depending on hidden neural states such as up and down states (in this case MPHs also allow to infer the up and down states, for instance through the generalized Viterbi algorithm), and left-to-right HMMs [40] can incorporate behavioral context in stereotyped motor actions such as birdsong. MPHs can bridge between data analysis and theories of neural function. Some theories of cortical function assume discrete modules of computation and representation [56], [57], for example synfire chains [47], [58] or, more generally, cell assemblies. In these theories, the role of neural activity does not only depend on the identity of the neuron but also on the (hidden) identity of modules the neuron belongs to at a certain time. The MPHs we developed to align stimulus and neural response are based on stimuli represented with continuous probability densities and neural activity represented with discrete probabilities. It is noteworthy that both fully continuous pair HMMs that align two continuous sequences and fully discrete pair HMMs also have interesting applications. For instance, we have shown previously that a fully continuous pair HMM can be used to align the songs of a juvenile bird to the song of the bird' s tutor in order to identify the parts of the song that were copied and the locations where insertions were made [59]. We have also demonstrated how fully discrete pair HMMs can be used to align spike trains [59]: by learning a discrete pair HMM on pairs of related spike trains, we obtained a “distance” measure between spike-trains, thereby generalizing state of the art spike train metrics [60]. MPHs are useful for both neural encoding and decoding. We presented algorithms for inferring neural responses and their probabilities given the stimulus (encoding). However, by symmetry of MPHs, the inference algorithms we designed can in principle be “inverted” to estimate the stimulus given neural activity (decoding) so that decoding and encoding of brain activity essentially have become the same problem. MPHs are based on classical hidden Markov models and learning and inference algorithms other than the EM algorithm are readily available. For instance, for model parameter estimation we could have used (much faster) Viterbi training [38] or we could have optimized criteria other than data likelihood [61]. Also, there exists a large variety of very powerful analytical and computational tools developed for classical hidden Markov models that can be adapted to MPHs [61]–[63]. We will make a code package for fitting MPHs available through our website (www. ini. ch/~skollmor).
All experiments were carried out in accordance with protocols approved by the Veterinary Office of the Canton of Zurich, Switzerland. We provide a short introduction to “normal” hidden Markov models and the associated terminology for readers unfamiliar with them. Consider two dies at a game of chance, one with equal probabilities for its six faces (fair die) and the other with unequal probabilities (loaded die). Suppose that their holders can exchange dies for one another without you knowing. Suppose furthermore that these die switches occur randomly. All you observe is the sequence of faces without knowing whether the fair or loaded die is in place: the identity of the die is hidden from you. Hidden Markov models (HMMs) account for exactly these kinds of situations involving hidden variables. In the die example we can use an HMM with two states, L (loaded) and F (fair), for the two dies. At any point in time the HMM is in one of the two states corresponding to the die that is in use. Associated with each of the two states are the probabilities for the different faces to come up. These emission probabilities are unknown and can be learned from observations (the distribution is uniform for the fair die and non-uniform for the loaded one). Transitions between states (dies) are governed by unknown transitions probabilities that model how likely the die holders switch dies at any time. For two states, transitions are modeled by an unknown 2 by 2 transition matrix that can also be learned from observations. An HMM can produce observations by randomly choosing transitions (die switches) and observations or emissions (faces that come up) which results in an observation sequence and an underlying hidden state sequence. However, HMMs are so useful because they can be applied in reverse: given an observation sequence, we can estimate good parameters (emission and transition probabilities) for the underlying HMM as well as the underlying hidden state sequence, which we never directly observed. In a classical HMM (applied to stimulus-response modeling) the time lag between the stimulus and the response is fixed and together stimulus and response probabilistically depend on some hidden (non-observed) variable with Markov dynamics. In an MPH, spike and stimuli also probabilistically depend on some hidden variable, but rather than being paired at a fixed time lag, spike and stimulus pairing is dynamic, governed by a probabilistic process. Note that MPHs are different from factorial hidden Markov models which employ a distributed state representation but model a single (possibly multidimensional) observation sequence [64]. In the following, we present a precise definition of the MPH architecture and its learning and inference algorithms. We denote the stimulus sequence by and the spiking response by where and are their respective durations (typically). are real vectors (e. g. sound spectrograms) and are integers (e. g. , number of spikes, typically in small time bins with zero or one spike). We denote a position in the combined stimulus-response alignment matrix as, Fig. 2c. The model has three types of hidden states: X-states, which model only the stimulus, R-States which model only the response, and M-States which jointly model stimulus and response (Fig. 2c). We denote the sets of these states by, , and. Additionally we define the union of states. We denote sequences of hidden states by with and use the notation to refer to the hidden state occupied at sequence position. Note that in general because all of stimulus, response, and hidden state sequences may be of different length. The parameters of the MPH are defined in the following. : Matrix of transition probabilities. denotes the probability of transiting from hidden state to hidden state: Initial probability of hidden state: Final probability of hidden state for: Emission probability density of the stimulus given hidden X-state for: Discrete emission probability distribution of the response given hidden R-state for: Mixed discrete-continuous emission probability of stimulus-response pair given hidden state As emission probability densities associated with X and M states we use multivariate Gaussians or mixtures of Gaussians, respectively: where is the weight of the kth mixture component, denotes the total number of mixture components (which may vary for different hidden states but this freedom is not reflected in our notation), and and denote Gaussian mean and covariance matrix of the th mixture component. For M states we keep track of one such density for each possible value of (distinct stimulus emission for each spiking state). In the following, we define algorithms for inference in MPHs. Some of them are generalizations of well-known algorithms for normal HMMs. To infer the spiking response for a given stimulus, we derive new algorithms. In the following we denote conditional probabilities of the form simply by, i. e. , for readability we will omit the dependence on model parameters. Assume that we have trained MPH model parameters on some data and now would like to apply the MPH to novel stimulus-response pairs. In a switching model (Fig. 2e), we would like to estimate the most likely hidden state sequence given the data to identify the switching events. In a flexible timing model (Fig. 2b) we would like to determine the optimal alignment between stimulus and response to estimate the jitter of individual spikes. In that latter case, the alignment consists of temporal offsets between stimulus and response on a moment-to-moment basis. The generalized Viterbi algorithm for MPHs can be applied in both situations to efficiently compute the most likely hidden state sequence for a given stimulus-response sequence: We apply an extension of the Viterbi algorithm for classical HMMs [38]. First, let be the probability of the most likely sequence that models the stimulus up to (and including) time, the response up to time, and that ends in hidden state. Additionally, for any state and sequence position, we keep track of the most likely precursor state in. and can be computed recursively (Table 1). A good way to visualize the generalized Viterbi algorithm is to think of it as filling up an alignment tensor (Fig. 2f). The final state of the most likely hidden state sequence is then given by and the complete state sequence can be obtained by iteratively back-tracking the most likely precursor states In many cases, we are interested in computing statistics over all possible sequences. For instance, to compute the probability of generating a sequence pair given a particular MPH (for example to compare different MPHs), we need to consider the overall probability of independent of the alignment. Hence we have to consider all possible hidden state sequences and not just the one with maximal likelihood. First, let be the probability of observing the stimulus up to (and including) time, the response up to time, and of ending in hidden state. The computation of is very similar to the computation of, except that the max operation is replaced with a summation (to take all hidden state sequences into account, Table 2). The backward algorithm is analogues to the forward algorithm. We present it here because it is an integral part of the EM algorithm for MPHs and the computation of posterior probabilities (see below). The backward probability is the probability of observing the stimulus from time to the end and the response from time to the end (excluding), beginning at position and in hidden state (. is computed recursively (Table 3). Assume we have trained an MPH on some data and want to determine the probability distribution over hidden states for a given stimulus-response pair and sequence position. Building on the definitions of and (Table 2 and 3), the posterior probability of being in hidden state at sequence position given sequence-pair can be expressed in terms of forward and backward probabilities: (8) Intuitively, the alignment kernel is a histogram of spike shifts over all possible state paths weighted by their respective probability. The alignment kernel is easily computed using posterior probabilities (Eq. 8) in M- and R-States at all sequence positions which fulfill: Negative lags in the alignment kernel imply that spikes occur before corresponding events in the stimulus, whereas positive lags imply that spikes occur thereafter. To train an MPH on a set of stimulus-response pairs, we apply a generalization of the EM algorithm. That algorithm is analogous to its normal HMM counterpart [38], [40]. In the expectation step, the and (Table 2 and 3) are used to compute the probability of each state at each sequence position, as well as the expected number of transitions between hidden state pairs. The model parameters are then re-estimated in such a way as to locally maximize the likelihood of the stimulus-response pair. For simplicity of notation we define as the probability of transiting from state to state at sequence position (: Based on, the new transition probabilities are given by Initial probabilities are updated similarly: The new discrete emission probabilities for R-States are given bywhere and. The update of emission density parameters for X states depends on the type of continuous probability distribution used. For Gaussian mixtures with mixture components, new means and covariance matrices for the components can be computed as follows. For simplicity, we first define, where and: The updated mixture weights, , the means, , and the covariance matrices are then computed as follows: The updates for M-states are analogous. To compute the updated parameters of the mixture associated with (where is the number of possible neural responses, i. e. is the maximum number of spikes per time bin), we sum only over those sequence positions that fulfill. Given an MPH that was trained on some stimulus-response pairs, we can predict spiking responses to novel stimuli. This is known as encoding. Conversely, we can reconstruct stimuli from spiking responses, known as decoding. In the following, we derive two encoding algorithms for MPHs. First, we show how to compute the most likely pair of hidden-state and neural response sequences, , for a given stimulus. This algorithm is an extension of the generalized Viterbi algorithm (Table 1). We only present the algorithm for encoding. By symmetry, a decoding algorithm can be derived analogously. Let again be the probability of the most likely hidden state sequence that models the stimulus up to time and the response up to time and ends in state. We want to compute a neural response such that is maximized, where denotes the most likely state path for the sequence pair (Table 1). This is accomplished by always choosing the instantaneous neural response, such that it maximizes the emission probability in the recursion equations (Table 4). As in the generalized Viterbi algorithm (Table 1), we keep track of the most likely precursor states in. Additionally, we store the emissions that maximize the first factor on the right hand side of the recursion equations as. We recover the most likely pair of hidden state sequence and neural response by considering the associated with the most likely state at that position (we assume that; generalization to unknown is possible, but irrelevant for our purposes). This encoding strategy yields a spike train which depends on the most likely hidden state sequence. Such dependence can be a problem if many pairs of hidden state sequences exist with similarly high probability. Also, another caveat is that this algorithm does not provide spiking probabilities. Ideally, we would like to account for all possible hidden state sequences and compute an overall spiking or response probability for each point in time. Such improvement can be done through an extension of the forward algorithm, presented next. Here we compute the probability distribution of the response at time given a stimulus sequence We can retrieve this probability as a posterior (using Eq. 8) after rewriting our model in the following way. With this reformulation, we can now easily express using sums over posterior probabilities of the and states: where denotes the posterior probability of hidden state in the rewritten model, and denote the sets of all ‘spiking’ and non-spiking R-states, respectively. Note that by construction is independent of the response. In this paper we always use this algorithm for inferring spiking probabilities in MPHs. Filling out the alignment tensor used to compute forward and backward probabilities (Fig. 2f) in a fully connected model requires computations and additional computations for emission probabilities in M and X states (as before, denotes the length of the sequence (stimulus), the length of the sequence (response); and is the number of hidden states). We usually reduce this complexity by limiting the allowed temporal offset between stimulus and response to a maximal lag set by a parameter. In that case, we compute only the part of the alignment tensor within a band of width around the diagonal. Hence, the complexity reduces to. In the EM algorithm, the computational complexity is. The MPHs we studied had mostly constrained parameters, in particular constrained covariance matrices and means. We have found that free covariance matrices tend to make the models prone to over fitting and slow down training as more iterations of the EM algorithm are required (for instance, the M-MPH discussed in the section on LNP equivalence reaches the optimum in one iteration. Using free covariance matrices, convergence is gradual and it takes many more steps for the likelihood change to drop below a predefined threshold). The EM algorithm only converges to local optima; we found that this problem can be alleviated by running the training several times from different initializations (compare Fig. 5D and the accompanying text). All experiments were carried out in accordance with protocols approved by the Veterinary Office of the Canton of Zurich, Switzerland. Data were collected from juvenile male zebra finches (60–92 days old). The electrophysiological procedures are explained in detail elsewhere [65]. Briefly, microdrives were implanted using methods previously described [65]. After each experiment, the brain was removed for histological examination of unstained slices to verify the location of reference lesions. Cells were recorded during singing. During recording sessions, birds were housed in a sound isolation chamber equipped with a microphone. Extracellular voltage traces were digitized at 33 kHz and recorded for offline spike sorting.
The brain computes using electrical discharges of nerve cells, so called spikes. Specific sensory stimuli, for instance, tones, often lead to specific spiking patterns. The same is true for behavior: specific motor actions are generated by specific spiking patterns. The relationship between neural activity and stimuli or motor actions can be difficult to infer, because of dynamic dependencies and hidden nonlinearities. For instance, in a freely behaving animal a neuron could exhibit variable levels of sensory and motor involvements depending on the state of the animal and on current motor plans—a situation that cannot be accounted for by many existing models. Here we present a new type of model that is specifically designed to cope with such changing regularities. We outline the mathematical framework and show, through computer simulations and application to recorded neural data, how MPHs can advance our understanding of stimulus-response relationships.
neuroscience
coding mechanisms
biology
computational neuroscience
Dynamic Alignment Models for Neural Coding