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![Covariance analysis using Gaussian (b) and spherical non-Gaussian (c) stimulus distributions. (a1) Prior eigenvalues. (a2) and (a3) Filters governing the firing probability in the time and frequency domains. (a4) Spike probability in the relevant space: P(spike|s) = (1 − exp{−[(k T 1 s/2.2) 2 + (k T 2 s/2.2) 2 ]}) 4. (b) Gaussian prior stimulus. (b1) Prior (gray) and spike-generating (red) stimuli. (b2) Eigenvalues of C s. The black line indicates the value 1. (b3) Eigenvectors e 1 and e 2 corresponding to the eigenvalues of matching colors in b2 and comparison with the filters k 1 and k 2. (b4) Comparison between the filters k 1 and k 2 and their projections k 1 and k 2 on the space generated by e 1 and e 2. (c) Same as above, for non-Gaussian prior stimuli. The stimuli belong to the surface of a 20-dimensional sphere with unit variance along each component. Number of analyzed spikes in each example: 5000](profile/Ines-Samengo/publication/229153344/figure/fig4/AS:667204592885771@1536085379657/Covariance-analysis-using-Gaussian-b-and-spherical-non-Gaussian-c-stimulus.png)
Covariance analysis using Gaussian (b) and spherical non-Gaussian (c) stimulus distributions. (a1) Prior eigenvalues. (a2) and (a3) Filters governing the firing probability in the time and frequency domains. (a4) Spike probability in the relevant space: P(spike|s) = (1 − exp{−[(k T 1 s/2.2) 2 + (k T 2 s/2.2) 2 ]}) 4. (b) Gaussian prior stimulus. (b1) Prior (gray) and spike-generating (red) stimuli. (b2) Eigenvalues of C s. The black line indicates the value 1. (b3) Eigenvectors e 1 and e 2 corresponding to the eigenvalues of matching colors in b2 and comparison with the filters k 1 and k 2. (b4) Comparison between the filters k 1 and k 2 and their projections k 1 and k 2 on the space generated by e 1 and e 2. (c) Same as above, for non-Gaussian prior stimuli. The stimuli belong to the surface of a 20-dimensional sphere with unit variance along each component. Number of analyzed spikes in each example: 5000
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The space of sensory stimuli is complex and high-dimensional. Yet, single neurons in sensory systems are typically affected by only a small subset of the vast space of all possible stimuli. A proper understanding of the input-output transformation represented by a given cell therefore requires the identification of the subset of stimuli that are re...
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... Gaussian and non-Gaussian stimuli In the example of Fig. 5, the difference between Gaussian and non-Gaussian prior stimuli is exemplified. Both ap- plied stimulus distributions are spherically symmetric, and the eigenvalues of their prior covariance matrices are all equal to 1 (panel (a1)). The relevant space is spanned by the filters k 1 and k 2 , and these two vectors differ in their shape ...
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... the eigenvalues of their prior covariance matrices are all equal to 1 (panel (a1)). The relevant space is spanned by the filters k 1 and k 2 , and these two vectors differ in their shape (panel (a2)) and frequency content (panel (a3)). The firing probability (panel (a4)) has rotational symmetry in the relevant space. When the stimulus is Gaussian (Fig. 5(b)), all irrelevant eigenval- ues cluster around unity (panel (b2)). In contrast, for 4 Covariance analysis can be carried out with (c) or without (d) subtracting the STA. (a) Spike probability in the relevant ...
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... both scenarios of Fig. 5, the two relevant stim- ulus directions are identified by the two outliers of the spectrum (panels (b2) and (c2)). Note that the obtained relevant eigenvectors and the original filters of the model do not match in a one-by-one fashion. The two pairs of vectors, however, span the same space, since each filter k m coincides with its ...
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... is, in fact, all that one can expect from STC analysis; in the expression of the firing probability, Eq. (5), the individual filters k m are not uniquely defined and could be exchanged for others that span the same relevant space, provided that the nonlinearity ϕ be appropriately adjusted. Thus, the expression of the firing probability used in Fig. 5 could ...
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... present two examples where the prior stimulus distribution is elliptic. The first one is a Gaussian stim- ulus, for which different dimensions are independent from one another. The second one is a hollow-ellipsoid- like stimulus distribution, where components are cou- pled. In both cases, we employ the same nonlinearity and relevant filters as in Fig. 5. Our aim is to compare the results obtained by diagonalizing C −1 p C s with those of C = C s − C p . As expected, the two methods are equivalent when the stimulus is Gaussian, but produce different results when applied to non-Gaussian elliptic stimulus ...
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... two relevant filters (panels (a2) and (a3)) and the spiking probability are identical to those used in previous examples (Figs. 5 and 6). In the present case, however, the prior stimulus has approximately constant variance throughout the frequency range covered by the two filters (panel (a3)). Thus, the relevant space is almost fully included in the subspace spanned by the short directions of the prior stimulus and is perpendic- ular to the two elongated directions. ...
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... the value that would be expected in the limit of infinite amounts of data. The question then arises whether the scatter ob- served in a given spectrum represents true differences in variance along relevant directions or instead results from statistical fluctuations along irrelevant directions. Figure 8 displays the spectra of the same model as in Fig. 5(c), but now varying the number of spikes included in the analysis. In panel (a1), only 23 spikes are employed, and there, it is not possible to determine by naked eye which eigenvalues belong to the degenerate baseline level and which are the outliers. In panel (a3), on the other hand, with 5,000 spikes, the task appears trivial. To see ...
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... like the resampling procedure that is based on shifting spike times, this analysis is performed in a nested fashion. The procedure is illustrated in Fig. 8(b) and (c) for the model used in Fig. 5(c) with a spherically symmetric stimulus distribution on the surface of a 20- dimensional sphere and two relevant directions. In the first round, all spike-triggered stimuli are rotated in the full N-dimensional stimulus space, and we test the null-hypothesis that there are no relevant directions, so the entire spike-triggered stimulus ...
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... inside the relevant space and if the associated irreducible space has dimension larger than 1, degeneracies also appear in K. Of course, P(s|spike) might have no symmetry in the relevant sub- space. But if, for example, ϕ(k 2 , then when the prior stimulus distribution is spherical, also the relevant directions have degenerate eigenvalues. In Fig. 5, for example, the relevant eigenvalues of C s are degener- ate. This is not the case in Fig. 4, where the firing probability is not symmetric. In Fig. 6, instead, the firing probability is indeed symmetric, but the prior stimulus is not. Thus the degeneracy with respect to the relevant eigenvectors of C −1 p C s is broken. In Fig. 7, ...
Citations
... STNMF is a type of spike-triggered analysis that aims at extracting spatial subunits from the structure of spike-eliciting stimulus segments under white-noise stimulation. Spike-triggered analyses have long been used for assessing receptive fields via computation of the spike-triggered average (STA; Bryant and Segundo 1976;Chichilnisky 2001;De Boer and Kuyper 1968) as well as for obtaining multiple (typically temporal) stimulus filters via spike-triggered covariance (STC) analysis (Cantrell et al. 2010;Fairhall et al. 2006;Gollisch and Meister 2008;Samengo and Gollisch 2013;Schwartz et al. 2006). As an extension of these approaches, STNMF can identify localized subunits, based on the statistical structure (e.g. ...
A standard circuit motif in sensory systems is the pooling of sensory information from an upstream neuronal layer. A downstream neuron thereby collects signals across different locations in stimulus space, which together compose the neuron’s receptive field. In addition, nonlinear transformations in the signal transfer between the layers give rise to functional subunits inside the receptive field. For ganglion cells in the vertebrate retina, for example, receptive field subunits are thought to correspond to presynaptic bipolar cells. Identifying the number and locations of subunits from the stimulus–response relationship of a recorded ganglion cell has been an ongoing challenge in order to characterize the retina’s functional circuitry and to build computational models that capture nonlinear signal pooling. Here we present a novel version of spike-triggered non-negative matrix factorization (STNMF), which can extract localized subunits in ganglion-cell receptive fields from recorded spiking responses under spatiotemporal white-noise stimulation. The method provides a more than 100-fold speed increase compared to a previous implementation, which can be harnessed for systematic screening of hyperparameters, such as sparsity regularization. We demonstrate the power and flexibility of this approach by analyzing populations of ganglion cells from salamander and primate retina. We find that subunits of midget as well as parasol ganglion cells in the marmoset retina form separate mosaics that tile visual space. Moreover, subunit mosaics show alignment with each other for ON and OFF midget as well as for ON and OFF parasol cells, indicating a spatial coordination of ON and OFF signals at the bipolar-cell level. Thus, STNMF can reveal organizational principles of signal transmission between successive neural layers, which are not easily accessible by other means.
... The mean (µ) was subtracted from the STA to show stimulus fluctuations around zero. For the spike-triggered covariance, the covariance matrix of the spike-triggered stimuli was formed based on the following equation(Samengo and Gollisch, 2013) ...
Neurons use action potentials, or spikes, to process information. Different aspects of spiking, such as its rate or timing, are used to encode information about different features of an input. But noise can influence how robustly information is represented in each coding scheme. Additionally, information can be lost if neural representations are not transmitted with high fidelity to downstream neurons. In both cases, properties of the input (its amplitude and kinetics) and properties of the neuron (its spike initiation mechanism and excitability) impact neural information processing.
In my thesis, I first investigated how axons are optimized to transmit spike-based representations. Using patch clamp electrophysiology combined with optogenetics, I showed that the axon of CA1 pyramidal neurons spikes transiently in response to sustained depolarization, in contrast to the soma and axon initial segment, which spike repetitively. These distinct spiking patterns are due to the differential expression of ion channels, supporting functional specialization of neuronal compartments. Specifically, low-threshold potassium channels (Kv1) cause the axon to behave as a high-pass filter, enabling high fidelity transmission of spike-based information so that the axon selectively responds to inputs with fast kinetics. Together with biophysical modeling, my findings demonstrate that spike initiation properties in each part of the neuron are well matched to the signals normally processed in that neuronal compartment.
I then investigated how background synaptic activity (noise) affects rate and temporal coding of vibrotactile stimuli. Using patch clamp electrophysiology and dynamic clamp in vitro, I found that layer 2/3 pyramidal neurons in primary somatosensory cortex spike intermittently to inputs repeated at frequencies perceived as vibration. The fraction of inputs evoking a spike varies with input amplitude, enabling firing rate to encode stimulus intensity. Despite being small in amplitude, inputs are abrupt in onset, which allows them to evoke precisely timed spikes, even under noisy conditions. Unreliable spiking allows noise to produce irregular skipping, enabling spike times (patterns) to encode stimulus frequency. The reliability and precision of spikes depend on input amplitude and kinetics, respectively. With the help of simulations, my results show that noise helps multiplexed rate and temporal coding.
... therefore only a few significant STC eigenvectors are generally used to indirectly capture the structure of an RGC' s RF [12,14,[21][22][23][24][25][26]. Such significant STC eigenvectors could be interpreted as additional RGC functional filters not found in the STA. ...
... Despite the wealth of studies using STC eigenvectors, a consensus interpretation seems to be lacking [12,14,[21][22][23][24][25][26]. For instance, Fairhall et al. [13] investigated the temporal structure of RGC RFs by STC analysis for rapidly changing but spatially uniform visual stimuli. ...
Retinal ganglion cells (RGCs), the retina's output neurons, encode visual information through spiking. The RGC receptive field (RF) represents the basic unit of visual information processing in the retina. RFs are commonly estimated using the spike-triggered average (STA), which is the average of the stimulus patterns to which a given RGC is sensitive. Whereas STA, based on the concept of the average, is simple and intuitive, it leaves more complex structures in the RFs undetected. Alternatively, spike-triggered covariance (STC) analysis provides information on second-order RF statistics. However, STC is computationally cumbersome and difficult to interpret. Thus, the objective of this study was to propose and validate a new computational method, called spike-triggered clustering (STCL), specific for multimodal RFs. Specifically, RFs were fit with a Gaussian mixture model, which provides the means and covariances of multiple RF clusters. The proposed method recovered bipolar stimulus patterns in the RFs of ON-OFF cells, while the STA identified only ON and OFF RGCs, and the remaining RGCs were labeled as unknown types. In contrast, our new STCL analysis distinguished ON-OFF RGCs from the ON, OFF, and unknown RGC types classified by STA. Thus, the proposed method enables us to include ON-OFF RGCs prior to retinal information analysis.
... Covariance analysis has long been used in neuroscience when analyzing electrophysiological recordings of the activity of single sensory neurons that are selective to specific stimulus features Bryant andSegundo (1976), de Ruyter van Steveninck andBialek (1988), Simoncelli et al. (2004), Samengo and Gollisch (2013). The physiological signal is typically discrete (spike or no spike) and the stimulus varies from trial to trial, testing the effect of a collection of features that are the candidate relevant dimensions to which the neuron might be sensitive to. ...
... The matrix C 0 can be taken to its diagonal form D by a coordinate transformation D = O t C 0 O, where O is an orthogonal matrix. Following Samengo and Gollisch (2013), in order to spot departures from the normal state, all sampled vectors s are transformed into vectors s = D − 1 /2 Os. The new coordinates are here referred to as the symmetric ones. ...
... This property is also evident in the range covered by the horizontal axes of Fig. 2b. Following Samengo and Gollisch (2013), in order to spot departures from the normal state, we make a coordinate transformation that turns the ellipsoidal distribution of the normal state into a spherical distribution, with unit variance in all directions. The resulting symmetric distribution is seen as a gray sphere in Fig. 3a, and the regions of space occupied by each crisis, as elongated (colored online) ellipsoids. ...
The amount of power in different frequency bands of the electroencephalogram (EEG) carries information about the behavioral state of a subject. Hence, neurologists treating epileptic patients monitor the temporal evolution of the different bands. We propose a covariance-based method to detect and characterize epileptic seizures operating on the band-filtered EEG signal. The algorithm is unsupervised and performs a principal component analysis of intra-cranial EEG recordings, detecting transient fluctuations of the power in each frequency band. Its simplicity makes it suitable for online implementation. Good sampling of the non-ictal periods is required, while no demands are imposed on the amount of data during ictal activity. We tested the method with 32 seizures registered in 5 patients. The area below the resulting receiver-operating characteristic curves was 87% for the detection of seizures and 91% for the detection of recruited electrodes. To identify the behaviorally relevant correlates of the physiological signal, we identified transient changes in the variance of each band that were correlated with the degree of loss of consciousness, the latter assessed by the so-called Consciousness Seizure Scale, summarizing the performance of the subject in a number of behavioral tests requested during seizures. We concluded that those crisis with maximal impairment of consciousness tended to exhibit an increase in variance approximately 40 s after seizure onset, with predominant power in the theta and alpha bands and reduced delta and beta activity.
... STC on the other hand makes use of the second moment to explore directions of differing variance in stimulus space [25]. Other groups have studied the theoretical properties of STC, its generalization to different stimulus distributions [35], and relation to Wiener / Volterra series [36], and demonstrated its benefits in previous studies on the V1 complex cells in macaques [37] and the H1 neurons in fly visual cortex [38]. As for the retinal system, STC analysis on RFs has been applied in salamander [39][40][41][42] and one monkey study [37], but not yet, to the best of our knowledge, to the mouse retina. ...
Retinal ganglion cells (RGCs) encode various spatiotemporal features of visual information into spiking patterns. The receptive field (RF) of each RGC is usually calculated by spike-triggered average (STA), which is fast and easy to understand, but limited to simple and unimodal RFs. As an alternative, spike-triggered covariance (STC) has been proposed to characterize more complex patterns in RFs. This study compares STA and STC for the characterization of RFs and demonstrates that STC has an advantage over STA for identifying novel spatiotemporal features of RFs in mouse RGCs. We first classified mouse RGCs into ON, OFF, and ON/OFF cells according to their response to full-field light stimulus, and then investigated the spatiotemporal patterns of RFs with random checkerboard stimulation, using both STA and STC analysis. We propose five sub-types (T1-T5) in the STC of mouse RGCs together with their physiological implications. In particular, the relatively slow biphasic pattern (T1) could be related to excitatory inputs from bipolar cells. The transient biphasic pattern (T2) allows one to characterize complex patterns in RFs of ON/OFF cells. The other patterns (T3-T5), which are contrasting, alternating, and monophasic patterns, could be related to inhibitory inputs from amacrine cells. Thus, combining STA and STC and considering the proposed sub-types unveil novel characteristics of RFs in the mouse retina and offer a more holistic understanding of the neural coding mechanisms of mouse RGCs.
... We found that one of these eigenvectors closely matched the STA, whereas the other corresponded to a motion trajectory in an orthogonal direction. For both eigenvectors, we then calculated conditional nonlinearities (Samengo and Gollisch, 2013) by selecting only those stimulus segments for which the stimulus projection onto the other eigenvector was small À0:5 < g j < 0:5 À Á and then computing nonlinearities in the same fashion as described above for the LN model. ...
Neurons in sensory systems are often tuned to particular stimulus features. During complex naturalistic stimulation, however, multiple features may simultaneously affect neuronal responses, which complicates the readout of individual features. To investigate feature representation under complex stimulation, we studied how direction-selective ganglion cells in salamander retina respond to texture motion where direction, velocity, and spatial pattern inside the receptive field continuously change. We found that the cells preserve their direction preference under this stimulation, yet their direction encoding becomes ambiguous due to simultaneous activation by luminance changes. The ambiguities can be resolved by considering populations of direction-selective cells with different preferred directions. This gives rise to synergistic motion decoding, yielding more information from the population than the summed information from single-cell responses. Strong positive response correlations between cells with different preferred directions amplify this synergy. Our results show how correlated population activity can enhance feature extraction in complex visual scenes. Direction-selective retinal ganglion cells encode visual motion but also respond to luminance changes. Kühn and Gollisch report that this dual-feature sensitivity creates ambiguities for decoding complex motion stimuli. Population decoding resolves these ambiguities and provides a synergistic motion readout.
... We found that one of these eigenvectors closely matched the STA, whereas the other corresponded to a motion trajectory in an orthogonal direction. For both eigenvectors, we then calculated conditional nonlinearities (Samengo and Gollisch, 2013) by selecting only those stimulus segments for which the stimulus projection onto the other eigenvector was small À0:5 < g j < 0:5 À Á and then computing nonlinearities in the same fashion as described above for the LN model. ...
... Covariance analysis has long been used in neuroscience when analyzing electrophysiological recordings of the activity of single sensory neurons that are selective to specific stimulus features [Bryant and Segundo, 1976, de Ruyter van Steveninck and Bialek, 1988, Simoncelli et al., 2004, Samengo and Gollisch, 2013. The physiological signal is typically discrete (spike or no spike) and the stimulus varies from trial to trial, testing the effect of a collection of features that are are the candidate relevant dimensions to which the neuron might be sensitive to. ...
... is an orthogonal matrix. Following [Samengo and Gollisch, 2013], in order to spot departures from the normal state, all sampled vectors s are transformed into vectors s ′ = D − 1 /2 Os. The new coordinates are here referred to as the symmetric ones. ...
... In other words, in normal circumstances, both the mean value and the variance along the low-frequency components are much larger than the mean and the variance along the high-frequency components. This property is also evident in the range covered by the horizontal axes of Fig. 2 B. Following [Samengo and Gollisch, 2013], in order to spot departures from the normal state, we make a coordinate transformation that turns the ellipsoidal distribution of the normal state into a spherical distribution, with unit variance in all directions. The resulting symmetric distribution is seen as a gray sphere in Fig. 3A, and the regions of space occupied by each crisis, as elongated (colored online) ellipsoids. ...
A large variety of methods has been proposed for automatic seizure detection in EEG signals. Those achieving maximal performance are based on machine-learning techniques, which require long training sessions with large labelled databases, and produce a verdict with no intuitive justification. As an alternative, we here explore an unsupervised algorithm applicable to intra-cranial EEG recordings that requires good sampling of the non-ictal periods, but imposes no demands on the amount of data during ictal activity. The algorithm analyses how the amount of power in each frequency band fluctuates ( an evaluation physicians are familiar with ) and can be implemented online. The analysis is performed electrode by electrode, thus providing the spatio-temporal sequence in which the affected regions are recruited into the crisis. We test it with 32 crisis registered in 5 patients, for which we also have the degree of loss of consciousness as determined from a behavioural analysis. The method can achieve a precision of 85% and a recall of 88% in the identification of seizures, and 77% and 88% in the identification of recruited electrodes. The intuitive nature of the analysis allows us to identify certain physiological features that are correlated with the degree of loss of consciousness. For example, epileptic crisis in which the variance of the power in the alpha band increases at around seizure onset are particularly likely to impair consciousness. We conclude that the PCA of the distribution of power in different frequency bands provides information both about the detection and the characterization of epileptic seizures.
... Fig. 2a in Rust et al, 2005;(McFarland et al., 2013;I. M. Park et al., 2013;Samengo & Gollisch, 2013;Schwartz et al., 2006;Vintch, Movshon, & Simoncelli, 2015)). Analyses inspired by the efficient coding hypothesis seek receptive field populations that efficiently encode natural stimuli also commonly report mismatched weight matrices (c.f. ...
To model the responses of neurons in the early visual system, at least three basic components are required: a receptive field, a normalization term, and a specification of encoding noise. Here, we examine how the receptive field, the normalization factor, and the encoding noise impact the model neuron responses to natural images and the signal-to-noise ratio for natural image discrimination. We show that when these components are modeled appropriately, the model neuron responses to natural stimuli are Gaussian distributed, scale-invariant, and very nearly maximize the signal-to-noise ratio for stimulus discrimination. We discuss the statistical models of natural stimuli that can account for these response statistics, and we show how some commonly used modeling practices may distort these results. Finally, we show that normalization can equalize important properties of neural response across different stimulus types. Specifically, narrowband (stimulus- and feature-specific) normalization causes model neurons to yield Gaussian-distributed responses to natural stimuli, 1/f noise stimuli, and white noise stimuli. The current work makes recommendations for best practices and it lays a foundation, grounded in the response statistics to natural stimuli, upon which principled models of more complex visual tasks can be built.
... The stimulus is seen as a continuous signal that flows into the area under study, as happens in ecological conditions. The underlying assumption is that, if the ensemble from which stimulus are drawn is similar to the natural environment, eventually, the relevant features will come up [18,19]. This approach has allowed scientists to discover many relevant stimulus features that had not been noticed using the classical strategy, as for example, multiple subfields in macaque V1 [20], or the correlated deflection of several whiskers in barrel cortex [21]. ...
In the study of the neural code, information-theoretical methods have the advantage of making no assumptions about the probabilistic mapping between stimuli and responses. In the sensory domain, several methods have been developed to quantify the amount of information encoded in neural activity, without necessarily identifying the specific stimulus or response features that instantiate the code. As a proof of concept, here we extend those methods to the encoding of kinematic information in a navigating rodent. We estimate the information encoded in two well-characterized codes, mediated by the firing rate of neurons, and by the phase-of-firing with respect to the theta-filtered local field potential. In addition, we also consider a novel code, mediated by the delta-filtered local field potential. We find that all three codes transmit significant amounts of kinematic information, and informative neurons tend to employ a combination of codes. Cells tend to encode conjunctions of kinematic features, so that most of the informative neurons fall outside the traditional cell types employed to classify spatially-selective units. We conclude that a broad perspective on the candidate stimulus and response features expands the repertoire of strategies with which kinematic information is encoded.
