Figure 1 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
![A pair of integrate-and-fire model neurons driven by partially shared and correlated presynaptic events.
A Each of the neurons and receives input from sources, of which are excitatory and are inhibitory. Both neurons share a fraction of their excitatory and inhibitory sources, whereas the fraction is independent for each neuron. Schematically represented spike trains on the left of the diagram show the excitatory part of the input; the inhibitory input is only indicated. A single source emits spike events with a firing rate , with marginal Poisson statistics. Correlated spiking is introduced in the common excitatory sources to both neurons. This pairwise correlation is realized by means of a multiple interaction process (MIP) [39] that yields a correlation coefficient of between any pairs of sources. In absence of a threshold, the summed input drives the membrane potential to a particular working point described by its mean and standard deviation and the correlation coefficient between the free membrane potentials , of both neurons. In presence of a threshold mean and variance of the membrane potential determine the output firing rate and their correlation in addition determines the output correlation , calculated by (2). B–E Direct simulation was performed using different values of common input fraction and four fixed values of input spike synchrony (as denoted in E). Each combination of and was simulated for seconds; gray coded data points show the average over independent realizations. Remaining parameters are given in Table 1. Solid lines in B and C are calculated as (5) and (6), respectively. In C, for convenience, is normalized by the common input fraction , so that in absence of synchrony (). E shows the output spike synchrony .](profile/Sonja-Gruen/publication/236207326/figure/fig2/AS:341139321311248@1458345360936/A-pair-of-integrate-and-fire-model-neurons-driven-by-partially-shared-and-correlated.png)
A pair of integrate-and-fire model neurons driven by partially shared and correlated presynaptic events. A Each of the neurons and receives input from sources, of which are excitatory and are inhibitory. Both neurons share a fraction of their excitatory and inhibitory sources, whereas the fraction is independent for each neuron. Schematically represented spike trains on the left of the diagram show the excitatory part of the input; the inhibitory input is only indicated. A single source emits spike events with a firing rate , with marginal Poisson statistics. Correlated spiking is introduced in the common excitatory sources to both neurons. This pairwise correlation is realized by means of a multiple interaction process (MIP) [39] that yields a correlation coefficient of between any pairs of sources. In absence of a threshold, the summed input drives the membrane potential to a particular working point described by its mean and standard deviation and the correlation coefficient between the free membrane potentials , of both neurons. In presence of a threshold mean and variance of the membrane potential determine the output firing rate and their correlation in addition determines the output correlation , calculated by (2). B–E Direct simulation was performed using different values of common input fraction and four fixed values of input spike synchrony (as denoted in E). Each combination of and was simulated for seconds; gray coded data points show the average over independent realizations. Remaining parameters are given in Table 1. Solid lines in B and C are calculated as (5) and (6), respectively. In C, for convenience, is normalized by the common input fraction , so that in absence of synchrony (). E shows the output spike synchrony .
Source publication
The functional significance of correlations between action potentials of neurons is still a matter of vivid debate. In particular, it is presently unclear how much synchrony is caused by afferent synchronized events and how much is intrinsic due to the connectivity structure of cortex. The available analytical approaches based on the diffusion appr...
Similar publications
Cortical networks, \textit{in-vitro} as well as \textit{in-vivo}, can
spontaneously generate a variety of collective dynamical events such as network
spikes, UP and DOWN states, global oscillations, and avalanches. Though each of
them have been variously recognized in previous works as expressions of the
excitability of the cortical tissue and the...
Citations
... [60][61][62][63]. A diverging slope of the correlation-transmission curve, shown here for binary neurons, has also been demonstrated for spiking neurons without reset [62] and for the LIF model [61,[64][65][66], suggesting that these model classes behave similarly with regard to the transition to chaos. ...
Neurons in the brain communicate with spikes, which are discrete events in time and value. Functional network models often employ rate units that are continuously coupled by analog signals. Is there a qualitative difference implied by these two forms of signaling? We develop a unified mean-field theory for large random networks to show that first- and second-order statistics in rate and binary networks are in fact identical if rate neurons receive the right amount of noise. Their response to presented stimuli, however, can be radically different. We quantify these differences by studying how nearby state trajectories evolve over time, asking to what extent the dynamics is chaotic. Chaos in the two models is found to be qualitatively different. In binary networks, we find a network-size-dependent transition to chaos and a chaotic submanifold whose dimensionality expands stereotypically with time, while rate networks with matched statistics are nonchaotic. Dimensionality expansion in chaotic binary networks aids classification in reservoir computing and optimal performance is reached within about a single activation per neuron; a fast mechanism for computation that we demonstrate also in spiking networks. A generalization of this mechanism extends to rate networks in their respective chaotic regimes.
... A neuron receiving synchronous synaptic inputs is more likely to emit a spike than asynchronously arriving inputs, as predicted by theory (Abeles 1982;König et al. 1996;Fries 2005;Schultze-Kraft et al. 2013) and shown in experiments (Ashida et al. 2016). This observation led to hypothesize that neurons synchronize their activities beyond pairs. ...
Temporally, precise correlations between simultaneously recorded neurons have been interpreted as signatures of cell assemblies, i.e., groups of neurons that form processing units. Evidence for this hypothesis was found on the level of pairwise correlations in simultaneous recordings of few neurons. Increasing the number of simultaneously recorded neurons increases the chances to detect cell assembly activity due to the larger sample size. Recent technological advances have enabled the recording of 100 or more neurons in parallel. However, these massively parallel spike train data require novel statistical tools to be analyzed for correlations, because they raise considerable combinatorial and multiple testing issues. Recently, various of such methods have started to develop. First approaches were based on population or pairwise measures of synchronization, and later led to methods for the detection of various types of higher-order synchronization and of spatio-temporal patterns. The latest techniques combine data mining with analysis of statistical significance. Here, we give a comparative overview of these methods, of their assumptions and of the types of correlations they can detect.
Electronic supplementary material
The online version of this article (10.1007/s00422-018-0755-0) contains supplementary material, which is available to authorized users.
... In primary visual cortex they appear in relation to saccades (eye movements) [11,12]. Another argument for the functional relevance of correlations is the robustness of signals represented by synchronous activity against noise [13]. Non-Gaussian distributions of membrane potentials of neurons indeed point towards the synchronized arrival of synaptic events [14,15]. ...
Pairwise maximum-entropy models have been used in neuroscience to predict the activity of neuronal populations, given only the time-averaged correlations of the neuron activities. This paper provides evidence that the pairwise model, applied to experimental recordings, would produce a bimodal distribution for the population-averaged activity, and for some population sizes the second mode would peak at high activities, that experimentally would be equivalent to 90% of the neuron population active within time-windows of few milliseconds. Several problems are connected with this bimodality: 1. The presence of the high-activity mode is unrealistic in view of observed neuronal activity and on neurobiological grounds. 2. Boltzmann learning becomes non-ergodic, hence the pairwise maximum-entropy distribution cannot be found: in fact, Boltzmann learning would produce an incorrect distribution; similarly, common variants of mean-field approximations also produce an incorrect distribution. 3. The Glauber dynamics associated with the model is unrealistically bistable and cannot be used to generate realistic surrogate data. This bimodality problem is first demonstrated for an experimental dataset from 159 neurons in the motor cortex of macaque monkey. Evidence is then provided that this problem affects typical neural recordings of population sizes of a couple of hundreds or more neurons. The cause of the bimodality problem is identified as the inability of standard maximum-entropy distributions with a uniform reference measure to model neuronal inhibition. To eliminate this problem a modified maximum-entropy model is presented, which reflects a basic effect of inhibition in the form of a simple but non-uniform reference measure. This model does not lead to unrealistic bimodalities, can be found with Boltzmann learning, and has an associated Glauber dynamics which incorporates a minimal asymmetric inhibition.
... The problem of analytically calculating the stationary distributions conditional on a spike from the exit current at the threshold is also discussed in the case of colored noise [29]. A method to deal with very strong input correlations (c ≈ 1) in a specific input model (correlated Poisson processes) was suggested by [31]. Our study further suggests a novel technique, extending [25], to solve 2D jump equations for leaky integrate-and-fire neurons by mapping it to a Markov chain. ...
... This expression is also valid for more general input statistic models that rely on a decomposition into statistically independent parts [31,33]. The corresponding amplitude distribution q(m, n) can cover cases of non-Poissonian spike trains with nontrivial higher-order statistics, e.g. ...
... The case study reported here covers the leaky integrate-and-fire (LIF) model with Poisson input spike trains. However, our method can be easily generalized to non-linear leak functions [39], conductance based synaptic inputs [40] and more complex input correlation models [31,33],. although we have to leave the details of such generalizations open. In this section we will explain how to select a 'good basis for expansion', and we will give numerical examples that demonstrate the power of the method. ...
The relative timing of action potentials in neurons recorded from local cortical networks often shows a non-trivial dependence, which is then quantified by cross-correlation functions. Theoretical models emphasize that such spike train correlations are an inevitable consequence of two neurons being part of the same network and sharing some synaptic input. For non-linear neuron models, however, explicit correlation functions are difficult to compute analytically, and perturbative methods work only for weak shared input. In order to treat strong correlations, we suggest here an alternative non-perturbative method. Specifically, we study the case of two leaky integrate-and-fire neurons with strong shared input. Correlation functions derived from simulated spike trains fit our theoretical predictions very accurately. Using our method, we computed the non-linear correlation transfer as well as correlation functions that are asymmetric due to inhomogeneous intrinsic parameters or unequal input.
... As such, it detects SSEs which cannot be explained on the basis of rate coding mechanisms, and thus arise from spike correlations on a shorter temporal scale. Rate correlation is understood as a conceptually different mechanism of information processing than spike synchrony (as for SSEs), because it corresponds to stochastic (probability <1) rather than reliable (probability 1) neuron activation [29,[41][42][43]. ...
With the ability to observe the activity from large numbers of neurons simultaneously using modern recording technologies, the chance to identify sub-networks involved in coordinated processing increases. Sequences of synchronous spike events (SSEs) constitute one type of such coordinated spiking that propagates activity in a temporally precise manner. The synfire chain was proposed as one potential model for such network processing. Previous work introduced a method for visualization of SSEs in massively parallel spike trains, based on an intersection matrix that contains in each entry the degree of overlap of active neurons in two corresponding time bins. Repeated SSEs are reflected in the matrix as diagonal structures of high overlap values. The method as such, however, leaves the task of identifying these diagonal structures to visual inspection rather than to a quantitative analysis. Here we present ASSET (Analysis of Sequences of Synchronous EvenTs), an improved, fully automated method which determines diagonal structures in the intersection matrix by a robust mathematical procedure. The method consists of a sequence of steps that i) assess which entries in the matrix potentially belong to a diagonal structure, ii) cluster these entries into individual diagonal structures and iii) determine the neurons composing the associated SSEs. We employ parallel point processes generated by stochastic simulations as test data to demonstrate the performance of the method under a wide range of realistic scenarios, including different types of non-stationarity of the spiking activity and different correlation structures. Finally, the ability of the method to discover SSEs is demonstrated on complex data from large network simulations with embedded synfire chains. Thus, ASSET represents an effective and efficient tool to analyze massively parallel spike data for temporal sequences of synchronous activity.
... However, the functions and consequences of correlations, and whether correlated input may carry any information have long been debated (Shadlen and Newsome, 1998;Panzeri et al., 1999;Salinas et al., 2001;Averbeck et al., 2006;Wolfe et al., 2010;Dipoppa and Gutkin, 2013). One of the key questions is how input correlations are processed and transmitted from a layer of neurons to the next (Shadlen and Newsome, 1998;Diesmann et al., 1999;Salinas and Sejnowski, 2000;Reyes, 2003;de la Rocha et al., 2007;Ostojic et al., 2009;Litwin-Kumar et al., 2011;Hong et al., 2012;Schultze-Kraft et al., 2013). ...
... Most previous work has studied the problem of correlation transfer by resorting to further approximations of single neuron dynamics and considering the pairwise correlation between two neurons receiving correlated inputs. A typical strategy is to use the diffusion approximation, mimicking the synaptic inputs by currents with Gaussian white noise (de la Rocha et al., 2007;Ostojic et al., 2009;Litwin-Kumar et al., 2011;Hong et al., 2012;Schultze-Kraft et al., 2013). This approach assumes that autocorrelations in the inputs are small. ...
... In order to understand how input features, e.g., the level of background activity or input synchrony, affect correlation transfer, Ostojic et al. (2009) and Schultze-Kraft et al. (2013) studied changes of the membrane potential distribution of a neuron, and hence its probability of firing, in response to an additional input spike; Rosenbaum and Josić (2011) studied the conditional probability of a neuron to fire given that the other neuron has just recently fired. These approaches make the assumption that an input spike can contribute to at most one output spike, which is also problematic when neurons with slow synapses are considered, as synaptic filtering by slow synapses may induce burst firing (Moreno-Bote and Parga, 2004). ...
Neurons communicate and transmit information predominantly through spikes. Given that experimentally observed neural spike trains in a variety of brain areas can be highly correlated, it is important to investigate how neurons process correlated inputs. Most previous work in this area studied the problem of correlation transfer analytically by making significant simplifications on neural dynamics. Temporal correlation between inputs that arises from synaptic filtering, for instance, is often ignored when assuming that an input spike can at most generate one output spike. Through numerical simulations of a pair of leaky integrate-and-fire (LIF) neurons receiving correlated inputs, we demonstrate that neurons in the presence of synaptic filtering by slow synapses exhibit strong output correlations. We then show that burst firing plays a central role in enhancing output correlations, which can explain the above-mentioned observation because synaptic filtering induces bursting. The observed changes of correlations are mostly on a long time scale. Our results suggest that other features affecting the prevalence of neural burst firing in biological neurons, e.g., adaptive spiking mechanisms, may play an important role in modulating the overall level of correlations in neural networks.
... Theoretical models have emphasized that such correlations are an inevitable consequence if two neurons are part of the same network and share some synaptic input [2]. However, for non-linear neuron models, correlation functions are difficult to compute explicitly, especially for low firing rates in the strongly correlated regime [3,4]. Previous analytical approaches have employed perturbation theory [5] to study pair correlations under the assumption of weak input correlation [6]. ...
... However, there is ample evidence of massive shared input for pairs of nearby neurons, resulting in strong correlations particularly of their membrane potentials [1]. A full theory of correlations, covering the case of both weak and strong shared input alike, demands non-perturbative methods that take non-linear effects into account [4]. In the work presented here, we suggest a non-perturbative solution to the corresponding two-dimensional Fokker-Planck equation to describe correlated integrate-and-fire neurons in any regime, with arbitrary precision. ...
... Strong correlations of membrane potentials were observed in nearby neurons of cortical networks [1], compatible with the high degree of shared input suggested from neuroanatomical studies. In the strongly correlated regime the correlation transfer function is non-linear [4,6] and the dynamics is quite sensitive to heterogeneities of the input and of the model parameters [8]. Recent experiments demonstrated that asymmetric correlation functions arise in neocortical neurons as well [8]. ...
Pairs of neurons in brain networks often share much of the input they receive from other neurons. Due to essential non-linearities of neuronal dynamics, the consequences for the correlation of the output spike trains are not well understood in the strongly correlated regime. Here we consider two leaky integrate-and-fire neurons with correlated white noise input. We analyze this scenario using a novel non-perturbative approach. Hence our treatment covers both weakly and strongly correlated dynamics, generalizing previous results based on linear response theory.
... London et al., 2010). Spike synchrony is understood as a conceptually different mechanism of information processing than rate (co-)modulation, because it corresponds to reliable (probability 1) rather than stochastic (probability < 1) neuron activation (De la Rocha et al., 2007;Shea-Brown et al., 2008;Staude et al., 2008;Schultze-Kraft et al., 2013). ...
... As such, it detects SSEs which cannot be explained on the basis of rate coding mechanisms, and thus arise from spike correlations on a shorter temporal scale. Rate correlation is understood as a conceptually different mechanism of information processing than spike synchrony (as for SSEs), because it corresponds to stochastic (probability < 1) rather than reliable (probability 1) neuron activation (De la Rocha et al., 2007;Shea-Brown et al., 2008;Staude et al., 2008;Schultze-Kraft et al., 2013). ...
... The detected SSEs cannot be explained on the basis of rate coding mechanisms, and thus arise from spike correlations on a shorter temporal scale. Rate correlation is understood as a conceptually different mechanism of information processing than spike synchrony (as for SSEs), because it corresponds to stochastic (probability < 1) rather than reliable (probability 1) neuron activation (De la Rocha et al., 2007;Shea-Brown et al., 2008;Staude et al., 2008;Schultze-Kraft et al., 2013). ...
The mechanisms by which the brain represents, processes and transmits information in terms of spiking activity are largely unknown. The coordination of spikes among neurons at millisecond temporal scale has been hypothesized as a mechanism of information coding. Spike synchronization is considered a signature of such cooperative dynamics. Experimental studies have found evidence of the occurrence of spike synchronization beyond chance level in relation to behavior. However, these findings were mostly limited to data from a small number of simultaneously recorded neurons, and were concluded on the basis of pairwise or low-order correlation analyses unable to conclude on the presence of neuronal interactions of any order.
Recent advances in electrophysiological technology enable recordings of hundreds of neurons in parallel, a number which is destined to grow. The increase in data dimensionality makes traditional methods for spike correlation analysis inapplicable due to the combinatorical explosion in the number of patterns, which raises severe computational and statistical problems. New tools for the analysis of big data are thus needed.
This essay delivers two novel methods for the analysis of spike synchronization in massively parallel spike trains. The first method, called SPADE (spike pattern detection and evaluation) is designed to perform an optimal search for synchronous spike patterns using efficient data mining techniques. The detected patterns are then tested for significance using a hierarchy of statistical tests that circumvents multiple testing issues.
We employ SPADE to investigate the occurrence of spike synchronization in the motor cortex of two macaque monkeys performing an instructed-delay reach-to-grasp task. The task consists of different epochs and was performed in different ways. By analyzing each epoch and trial type separately, we are able to associate the emergence of individual patterns of synchronous spikes to specific behaviors, and we investigate the spatial organization of synchronous spiking neurons in motor cortex. Our analysis reveals groups of neurons,
aligned along a preferential medio-lateral orientation, which selectively synchronize their activity in relation to behavior.
We finally introduce a second statistical method, called ASSET (analysis of sequences of synchronous events), that searches for temporal sequences of synchronous spike patterns. Synchrony propagation has been long suggested as a plausible mechanism of information coding and transmission, successfully implemented for instance in the synfire chain model in a manner compatible with electrophysiological constraints. Here we provide, to the best of our knowledge, the first statistical tool to detect sequential synchronous events in massively parallel spike trains.
In summary, this work delivers two novel powerful statistical tools, SPADE and ASSET, to analyze spike synchrony and synchrony propagation in parallel spike trains. These methods are designed for application to large data from hundreds or thousands of simultaneous recordings, and cope with typical features of real data, such as non-stationarity of firing rates in time and across neurons or regularity in the spiking activity. The application of SPADE to motor cortex data from two macaque monkeys performing a reach-to-grasp task allows us to shed new light on the spatial and temporal organization of spike synchronization in the cerebral cortex in relation to behavior.
... For a process r μ (t) that is not bound by zero this is not strictly fulfilled. Hence, ensemble averages over the spike train will contain correction terms that are proportional to the square root of the probability that r μ (t) is smaller than zero, which we calculated in Appendix A in Eq. (52). Consequently, using Eq. ...
... There are a number of studies that explored theoretically the case of weakly correlated neurons and employed perturbation methods to relate output spike train correlations to input correlations [46][47][48][49][50][51][52]. In this paper, we have considered the opposite limit of strongly correlated spike trains that are only weakly decorrelated due to intrinsic noise sources. ...
We study a population of spiking neurons which are subject to independent noise processes and a strong common time-dependent input. We show that the response of output spikes to independent noise shapes information transmission of such populations even when information transmission properties of single neurons are left unchanged. In particular, we consider two Poisson models in which independent noise either (i) adds and deletes spikes (AD model) or (ii) shifts spike times (STS model). We show that in both models suprathreshold stochastic resonance (SSR) can be observed, where the information transmitted by a neural population is increased with addition of independent noise. In the AD model, the presence of the SSR effect is robust and independent of the population size or the noise spectral statistics. In the STS model, the information transmission properties of the population are determined by the spectral statistics of the noise, leading to a strongly increased effect of SSR in some regimes, or an absence of SSR in others. Furthermore, we observe a high-pass filtering of information in the STS model that is absent in the AD model. We quantify information transmission by means of the lower bound on the mutual information rate and the spectral coherence function. To this end, we derive the signal–output cross-spectrum, the output power spectrum, and the cross-spectrum of two spike trains for both models analytically.
... Similarly, pacemaker cells exhibit dominance over myocytes at an optimal gap junction conductance [46]; where high coupling leads to arrhythmias and low coupling leads to poor synchronization [47]. Neurons also display intrinsic oscillatory behavior, and the effects of coupling and presence of inhibitory and excitatory neurons on synchronization and phase modulation is an active area of research [48][49][50][51][52][53]. Critical dynamics have been described theoretically to emerge from excitatory and inhibitory units in neuronal networks [54], and a computational study which introduced 'contrarian elements' into neural coupled oscillator networks found that a similar threshold of 15% suppressed global dynamics [49]. ...
The pancreatic islets of Langerhans are multicellular micro-organs integral to maintaining glucose homeostasis through secretion of the hormone insulin. β-cells within the islet exist as a highly coupled electrical network which coordinates electrical activity and insulin release at high glucose, but leads to global suppression at basal glucose. Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated. Previous work has suggested that a small threshold of quiescent cells is able to suppress the entire network. By modeling the islet as a Boolean network, we predicted a phase-transition between globally active and inactive states would emerge near this threshold number of cells, indicative of critical behavior. This was tested using islets with an inducible-expression mutation which renders defined numbers of cells electrically inactive, together with pharmacological modulation of electrical activity. This was combined with real-time imaging of intracellular free-calcium activity [Ca2+]i and measurement of physiological parameters in mice. As the number of inexcitable cells was increased beyond ∼15%, a phase-transition in islet activity occurred, switching from globally active wild-type behavior to global quiescence. This phase-transition was also seen in insulin secretion and blood glucose, indicating physiological impact. This behavior was reproduced in a multicellular dynamical model suggesting critical behavior in the islet may obey general properties of coupled heterogeneous networks. This study represents the first detailed explanation for how the islet facilitates inhibitory activity in spite of a heterogeneous cell population, as well as the role this plays in diabetes and its reversal. We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range. These findings also give new insight into emergent multicellular dynamics in general which are applicable to many coupled physiological systems, specifically where inhibitory dynamics result from coupled networks.
