Resting state fMRI signals in mammals exhibit rich dynamics on a fast, frame-by-frame timescale of seconds, including the robust emergence of recurring fMRI co-activation patterns (CAPs). To understand how such dynamics emerges from the underlying anatomical cortico-cortical connectivity, we developed a whole-cortex model of resting-state fMRI signals in the mouse. Our model implemented neural input-output nonlinearities and excitatory-inhibitory interactions within cortical regions, as well as directed anatomical connectivity between regions inferred from the Allen mouse brain atlas. We found that, even if the model parameters were fitted to explain static properties of fMRI signals on the timescale of minutes, the model generated rich frame-by-frame attractor dynamics, with multiple stationary and oscillatory attractors. Guided by these theoretical predictions, we found that empirical mouse fMRI time series exhibited analogous signatures of attractor dynamics, and that model attractors recapitulated the topographical organization of empirical fMRI CAPs. The model established key relationships between attractor dynamics, CAPs and features of the directed cortico-cortical intra- and inter-hemispheric anatomical connectivity. Specifically, we found that neglecting fiber directionality severely affected the number of model's attractors and their ability to explain CAPs. Furthermore, modifying inter-hemispheric anatomical connectivity strength by decreasing or increasing it from the value of real mouse anatomical data, resulted in fewer attractors generated by cortico-cortical interactions and reduced non-homotopic features of the attractors generated by the model, which were important for better predicting empirical CAPs. These results offer novel theoretical insight into the dynamic organization of resting state fMRI in the mouse brain and suggest that the frame-wise BOLD activity captured by CAPs reflects an emerging property of cortical dynamics resulting from directed cortico-cortical interactions.

Information theory has deeply influenced the conceptualization of brain information processing and is a mainstream framework for analyzing how neural networks in the brain process information to generate behavior. Information theory tools have been initially conceived and used to study how information about sensory variables is encoded by the activity of small neural populations. However, recent multivariate information theoretic advances have enabled addressing how information is exchanged across areas and used to inform behavior. Moreover, its integration with dimensionality-reduction techniques has enabled addressing information encoding and communication by the activity of large neural populations or many brain areas, as recorded by multichannel activity measurements in functional imaging and electrophysiology. Here, we provide a Multivariate Information in Neuroscience Toolbox (MINT) that combines these new methods with statistical tools for robust estimation from limited-size empirical datasets. We demonstrate the capabilities of MINT by applying it to both simulated and real neural data recorded with electrophysiology or calcium imaging, but all MINT functions are equally applicable to other brain-activity measurement modalities. We highlight the synergistic opportunities that combining its methods afford for reverse engineering of specific information processing and flow between neural populations or areas, and for discovering how information processing functions emerge from interactions between neurons or areas. MINT works on Linux, Windows and macOS operating systems, is written in MATLAB (requires MATLAB version 2018b or newer) and depends on 4 native MATLAB toolboxes. The calculation of one possible way to compute information redundancy requires the installation and compilation of C files (made available by us also as pre-compiled files). MINT is freely available at https://github.com/panzerilab/MINT with DOI doi.org/10.5281/zenodo.13998526 and operates under a GNU GPLv3 license.

Quantifying the amount, content and direction of communication between brain regions is key to understanding brain function. Traditional methods to analyze brain activity based on the Wiener-Granger causality principle quantify the overall information propagated by neural activity between simultaneously recorded brain regions, but do not reveal the information flow about specific features of interest (such as sensory stimuli). Here, we develop a new information theoretic measure termed Feature-specific Information Transfer (FIT), quantifying how much information about a specific feature flows between two regions. FIT merges the Wiener-Granger causality principle with information-content specificity. We first derive FIT and prove analytically its key properties. We then illustrate and test them with simulations of neural activity, demonstrating that FIT identifies, within the total information propagated between regions, the information that is transmitted about specific features. We then analyze three neural datasets obtained with different recording methods, magneto- and electro-encephalography, and spiking activity, to demonstrate the ability of FIT to uncover the content and direction of information flow between brain regions beyond what can be discerned with traditional analytical methods. FIT can improve our understanding of how brain regions communicate by uncovering previously unaddressed feature-specific information flow.

Understanding how the dynamics of neural networks is shaped by the computations they perform is a fundamental question in neuroscience. Recently, the framework of efficient coding proposed a theory of how spiking neural networks can compute low-dimensional stimulus signals with high efficiency. Efficient spiking networks are based on time-dependent minimization of a loss function related to information coding with spikes. To inform the understanding of the function and dynamics of biological networks in the brain, however, the mathematical models have to be informed by biology and obey the same constraints as biological networks. Currently, spiking network models of efficient coding have been extended to include some features of biological plausibility, such as architectures with excitatory and inhibitory neurons. However, biological realism of efficient coding theories is still limited to simple cases and does not include single neuron and network properties that are known to be key in biological circuits. Here, we revisit the theory of efficient coding with spikes to develop spiking neural networks that are closer to biological circuits. Namely, we find a biologically plausible spiking model realizing efficient coding in the case of a generalized leaky integrate-and-fire network with excitatory and inhibitory units, equipped with fast and slow synaptic currents, local homeostatic currents such as spike-triggered adaptation, hyperpolarization-activated rebound current, heterogeneous firing thresholds and resets, heterogeneous postsynaptic potentials, and structured, low-rank connectivity. We show how the rank of E-E connectivity matrix shapes network responses.

Sensory information is conveyed by populations of neurons, and coding strategies cannot always be deduced when considering individual neurons. Moreover, information coding depends on the number of neurons available and on the composition of the population when multiple classes with different response properties are available. Here, we study population coding in human tactile afferents by employing a recently developed simulator of mechanoreceptor firing activity. First, we highlight the interplay of afferents within each class. We demonstrate that the optimal afferent density to convey maximal information depends on both the tactile feature under consideration and the afferent class. Second, we find that information is spread across different classes for all tactile features and that each class encodes both redundant and complementary information with respect to the other afferent classes. Specifically, combining information from multiple afferent classes improves information transmission and is often more efficient than increasing the density of afferents from the same class. Finally, we examine the importance of temporal and spatial contributions, respectively, to the joint spatiotemporal code. On average, destroying temporal information is more destructive than removing spatial information, but the importance of either depends on the stimulus feature analyzed. Overall, our results suggest that both optimal afferent innervation densities and the composition of the population depend in complex ways on the tactile features in question, potentially accounting for the variety in which tactile peripheral populations are assembled in different regions across the body.

The electroencephalogram (EEG) is a major tool for non-invasively studying brain function and dysfunction. Comparing experimentally recorded EEGs with neural network models is important to better interpret EEGs in terms of neural mechanisms. Most current neural network models use networks of simple point neurons. They capture important properties of cortical dynamics, and are numerically or analytically tractable. However, point neurons cannot generate an EEG, as EEG generation requires spatially separated transmembrane currents. Here, we explored how to compute an accurate approximation of a rodent's EEG with quantities defined in point-neuron network models. We constructed different approximations (or proxies) of the EEG signal that can be computed from networks of leaky integrate-and-fire (LIF) point neurons, such as firing rates, membrane potentials, and combinations of synaptic currents. We then evaluated how well each proxy reconstructed a ground-truth EEG obtained when the synaptic currents of the LIF model network were fed into a three-dimensional network model of multicompartmental neurons with realistic morphologies. Proxies based on linear combinations of AMPA and GABA currents performed better than proxies based on firing rates or membrane potentials. A new class of proxies, based on an optimized linear combination of time-shifted AMPA and GABA currents, provided the most accurate estimate of the EEG over a wide range of network states. The new linear proxies explained 85-95% of the variance of the ground-truth EEG for a wide range of network configurations including different cell morphologies, distributions of presynaptic inputs, positions of the recording electrode, and spatial extensions of the network. Non-linear EEG proxies using a convolutional neural network (CNN) on synaptic currents increased proxy performance by a further 2-8%. Our proxies can be used to easily calculate a biologically realistic EEG signal directly from point-neuron simulations thus facilitating a quantitative comparison between computational models and experimental EEG recordings.

A fundamental and recurrent question in systems neuroscience is that of assessing what variables are encoded by a given population of neurons. Such assessments are often challenging because neurons in one brain area may encode multiple variables, and because neuronal representations might be categorical or non-categorical. These issues are particularly pertinent to the representation of decision variables in the orbitofrontal cortex (OFC)-an area implicated in economic choices. Here we present a new algorithm to assess whether a neuronal representation is categorical or non-categorical, and to identify the encoded variables if the representation is indeed categorical. The algorithm is based on two clustering procedures, one variable-independent and the other variable-based. The two partitions are then compared through adjusted mutual information. The present algorithm overcomes limitations of previous approaches and is widely applicable. We tested the algorithm on synthetic data and then used it to examine neuronal data recorded in the primate OFC during economic decisions. Confirming previous assessments, we found the neuronal representation in OFC to be categorical in nature. We also found that neurons in this area encode the value of individual offers, the binary choice outcome and the chosen value. In other words, during economic choice, neurons in the primate OFC encode decision variables in a categorical way.

Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

Determining how much of the sensory information carried by a neural code contributes to behavioral performance is key to understand sensory function and neural information flow. However, there are as yet no analytical tools to compute this information that lies at the intersection between sensory coding and behavioral readout. Here we develop a novel measure, termed the information-theoretic intersection information $\III(S;R;C)$, that quantifies how much of the sensory information carried by a neural response $R$ is used for behavior during perceptual discrimination tasks. Building on the Partial Information Decomposition framework, we define $\III(S;R;C)$ as the part of the mutual information between the stimulus $S$ and the response $R$ that also informs the consequent behavioral choice $C$. We compute $\III(S;R;C)$ in the analysis of two experimental cortical datasets, to show how this measure can be used to compare quantitatively the contributions of spike timing and spike rates to task performance, and to identify brain areas or neural populations that specifically transform sensory information into choice.

Concurrent measurements of neural activity at multiple scales, sometimes performed with multimodal techniques, become increasingly important for studying brain function. However, statistical methods for their concurrent analysis are currently lacking. Here we introduce such techniques in a framework based on vine copulas with mixed margins to construct multivariate stochastic models. These models can describe detailed mixed interactions between discrete variables such as neural spike counts, and continuous variables such as local field potentials. We propose efficient methods for likelihood calculation, inference, sampling and mutual information estimation within this framework. We test our methods on simulated data and demonstrate applicability on mixed data generated by a biologically realistic neural network. Our methods hold the promise to considerably improve statistical analysis of neural data recorded simultaneously at different scales.

Mean-field approximations are a powerful tool for studying large neural networks. However, they do not describe well the behavior of networks composed of a small number of neurons. In this case, major differences between the mean-field approximation and the real behavior of the network can arise. Yet, many interesting problems in neuroscience involve the study of mesoscopic networks composed of a few tens of neurons. Nonetheless, mathematical methods that correctly describe networks of small size are still rare, and this prevents us to make progress in understanding neural dynamics at these intermediate scales. Here we develop a novel systematic analysis of the dynamics of arbitrarily small networks composed of homogeneous populations of excitatory and inhibitory firing-rate neurons. We study the local bifurcations of their neural activity with an approach that is largely analytically tractable, and we numerically determine the global bifurcations. We find that for strong inhibition these networks give rise to very complex dynamics, caused by the formation of multiple branching solutions of the neural dynamics equations that emerge through spontaneous symmetry-breaking. This qualitative change of the neural dynamics is a finite-size effect of the network, that reveals qualitative and previously unexplored differences between mesoscopic cortical circuits and their mean-field approximation. The most important consequence of spontaneous symmetry-breaking is the ability of mesoscopic networks to regulate their degree of functional heterogeneity, which is thought to help reducing the detrimental effect of noise correlations on cortical information processing.

Advances in neuronal recording techniques are leading to ever larger numbers of simultaneously monitored neurons. This poses the important analytical challenge of how to capture compactly all sensory information that neural population codes carry in their spatial dimension (differences in stimulus tuning across neurons at different locations), in their temporal dimension (temporal neural response variations), or in their combination (temporally coordinated neural population firing). Here we investigate the utility of tensor factorizations of population spike trains along space and time. These factorizations decompose a dataset of single-trial population spike trains into spatial firing patterns (combinations of neurons firing together), temporal firing patterns (temporal activation of these groups of neurons) and trial-dependent activation coefficients (strength of recruitment of such neural patterns on each trial). We validated various factorization methods on simulated data and on populations of ganglion cells simultaneously recorded in the salamander retina. We found that single-trial tensor space-by-time decompositions provided low-dimensional data-robust representations of spike trains that capture efficiently both their spatial and temporal information about sensory stimuli. Tensor decompositions with orthogonality constraints were the most efficient in extracting sensory information, whereas non-negative tensor decompositions worked well even on non-independent and overlapping spike patterns, and retrieved informative firing patterns expressed by the same population in response to novel stimuli. Our method showed that populations of retinal ganglion cells carried information in their spike timing on the ten-milliseconds-scale about spatial details of natural images. This information could not be recovered from the spike counts of these cells. First-spike latencies carried the majority of information provided by the whole spike train about fine-scale image features, and supplied almost as much information about coarse natural image features as firing rates. Together, these results highlight the importance of spike timing, and particularly of first-spike latencies, in retinal coding.

Event-driven vision sensors have the potential to support a new generation of efficient and robust robots. This requires the development of a new computational framework that exploits not only the spatial information, like in the traditional frame-based approach, but also the temporal content of the sensory data. We propose a method for unsupervised learning of filters for the processing of the visual signal from event-driven sensors. This method exploits the temporal coincidence of events generated by each object in a spatial location of the visual field. The approach is based on a modification of Spike Timing Dependent Plasticity that takes into account the specific implementation on the robot and the characteristics of the used sensor. It gives rise to oriented spatial filters that are very similar to the receptive fields observed in the primary visual cortex and traditionally used in bio-inspired hierarchical structures for object recognition, as well as to novel curved spatial structures. Using mutual information measure we provide a quantitative evidence that such curved spatial filters provide more information than equivalent oriented Gabor filters and can be an important aspect for object recognition in robotic applications.

This letter introduces a study to precisely measure what an increase in spike timing precision can add to spike-driven pattern recognition algorithms. The concept of generating spikes from images by converting gray levels into spike timings is currently at the basis of almost every spike-based modeling of biological visual systems. The use of images naturally leads to generating incorrect artificial and redundant spike timings and, more important, also contradicts biological findings indicating that visual processing is massively parallel, asynchronous with high temporal resolution. A new concept for acquiring visual information through pixel-individual asynchronous level-crossing sampling has been proposed in a recent generation of asynchronous neuromorphic visual sensors. Unlike conventional cameras, these sensors acquire data not at fixed points in time for the entire array but at fixed amplitude changes of their input, resulting optimally sparse in space and time-pixel individually and precisely timed only if new, (previously unknown) information is available (event based). This letter uses the high temporal resolution spiking output of neuromorphic event-based visual sensors to show that lowering time precision degrades performance on several recognition tasks specifically when reaching the conventional range of machine vision acquisition frequencies (30-60 Hz). The use of information theory to characterize separability between classes for each temporal resolution shows that high temporal acquisition provides up to 70% more information that conventional spikes generated from frame-based acquisition as used in standard artificial vision, thus drastically increasing the separability between classes of objects. Experiments on real data show that the amount of information loss is correlated with temporal precision. Our information-theoretic study highlights the potentials of neuromorphic asynchronous visual sensors for both practical applications and theoretical investigations. Moreover, it suggests that representing visual information as a precise sequence of spike times as reported in the retina offers considerable advantages for neuro-inspired visual computations.

Leaky integrate-and-fire (LIF) network models are commonly used to study how the spiking dynamics of neural networks changes with stimuli, tasks or dynamic network states. However, neurophysiological studies in vivo often rather measure the mass activity of neuronal microcircuits with the local field potential (LFP). Given that LFPs are generated by spatially separated currents across the neuronal membrane, they cannot be computed directly from quantities defined in models of point-like LIF neurons. Here, we explore the best approximation for predicting the LFP based on standard output from point-neuron LIF networks. To search for this best "LFP proxy", we compared LFP predictions from candidate proxies based on LIF network output (e.g, firing rates, membrane potentials, synaptic currents) with "ground-truth" LFP obtained when the LIF network synaptic input currents were injected into an analogous three-dimensional (3D) network model of multi-compartmental neurons with realistic morphology, spatial distributions of somata and synapses. We found that a specific fixed linear combination of the LIF synaptic currents provided an accurate LFP proxy, accounting for most of the variance of the LFP time course observed in the 3D network for all recording locations. This proxy performed well over a broad set of conditions, including substantial variations of the neuronal morphologies. Our results provide a simple formula for estimating the time course of the LFP from LIF network simulations in cases where a single pyramidal population dominates the LFP generation, and thereby facilitate quantitative comparison between computational models and experimental LFP recordings in vivo.

While sensory neurons carry behaviorally relevant information in responses that often extend over hundreds of milliseconds, the key units of neural information likely consist of much shorter and temporally precise spike patterns. The mechanisms and temporal reference frames by which sensory networks partition responses into these shorter units of information remain unknown. One hypothesis holds that slow oscillations provide a network-intrinsic reference to temporally partitioned spike trains without exploiting the millisecond-precise alignment of spikes to sensory stimuli. We tested this hypothesis on neural responses recorded in visual and auditory cortices of macaque monkeys in response to natural stimuli. Comparing different schemes for response partitioning revealed that theta band oscillations provide a temporal reference that permits extracting significantly more information than can be obtained from spike counts, and sometimes almost as much information as obtained by partitioning spike trains using precisely stimulus-locked time bins. We further tested the robustness of these partitioning schemes to temporal uncertainty in the decoding process and to noise in the sensory input. This revealed that partitioning using an oscillatory reference provides greater robustness than partitioning using precisely stimulus-locked time bins. Overall, these results provide a computational proof of concept for the hypothesis that slow rhythmic network activity may serve as internal reference frame for information coding in sensory cortices and they foster the notion that slow oscillations serve as key elements for the computations underlying perception.

Progress in decoding neural signals has enabled the development of interfaces that translate cortical brain activities into commands for operating robotic arms and other devices. The electrical stimulation of sensory areas provides a means to create artificial sensory information about the state of a device. Taken together, neural activity recording and microstimulation techniques allow us to embed a portion of the central nervous system within a closed-loop system, whose behavior emerges from the combined dynamical properties of its neural and artificial components. In this study we asked if it is possible to concurrently regulate this bidirectional brain-machine interaction so as to shape a desired dynamical behavior of the combined system. To this end, we followed a well-known biological pathway. In vertebrates, the communications between brain and limb mechanics are mediated by the spinal cord, which combines brain instructions with sensory information and organizes coordinated patterns of muscle forces driving the limbs along dynamically stable trajectories. We report the creation and testing of the first neural interface that emulates this sensory-motor interaction. The interface organizes a bidirectional communication between sensory and motor areas of the brain of anaesthetized rats and an external dynamical object with programmable properties. The system includes (a) a motor interface decoding signals from a motor cortical area, and (b) a sensory interface encoding the state of the external object into electrical stimuli to a somatosensory area. The interactions between brain activities and the state of the external object generate a family of trajectories converging upon a selected equilibrium point from arbitrary starting locations. Thus, the bidirectional interface establishes the possibility to specify not only a particular movement trajectory but an entire family of motions, which includes the prescribed reactions to unexpected perturbations.

This paper describes a topological clustering of synchronous spike trains recorded in rat somatosensory cortex in response to sinusoidal vibrissal stimulations characterized by different frequencies and amplitudes. Discrete spike trains are first interpreted as continuous synchronous activities by a smoothing filter such as causal exponential function. Then clustering is performed using the self-organizing map, which yields topologically ordered clusters of responses with respect to the stimuli. The grouping is formed mainly along the product of amplitude and frequency of the stimuli. This result coincides with the result obtained previously using mutual information analysis on the same data set. That is, the response is proportional in logarithm to the energy of the vibration. It suggests that such clustering can naturally find underlying stimulus-response patterns and it also seems to associate the spike-count based mutual information decoding with temporal patterns of the neuronal activities. The study also shows that causal decaying exponential kernel is better than noncausal Gaussian kernel in interpreting the discrete spike trains into continues ones and produces better clusters.

Recordings of local field potentials (LFPs) reveal that the sensory cortex displays rhythmic activity and fluctuations over a wide range of frequencies and amplitudes. Yet, the role of this kind of activity in encoding sensory information remains largely unknown. To understand the rules of translation between the structure of sensory stimuli and the fluctuations of cortical responses, we simulated a sparsely connected network of excitatory and inhibitory neurons modeling a local cortical population, and we determined how the LFPs generated by the network encode information about input stimuli. We first considered simple static and periodic stimuli and then naturalistic input stimuli based on electrophysiological recordings from the thalamus of anesthetized monkeys watching natural movie scenes. We found that the simulated network produced stimulus-related LFP changes that were in striking agreement with the LFPs obtained from the primary visual cortex. Moreover, our results demonstrate that the network encoded static input spike rates into gamma-range oscillations generated by inhibitory-excitatory neural interactions and encoded slow dynamic features of the input into slow LFP fluctuations mediated by stimulus-neural interactions. The model cortical network processed dynamic stimuli with naturalistic temporal structure by using low and high response frequencies as independent communication channels, again in agreement with recent reports from visual cortex responses to naturalistic movies. One potential function of this frequency decomposition into independent information channels operated by the cortical network may be that of enhancing the capacity of the cortical column to encode our complex sensory environment.

The estimation of the information carried by spike times is crucial for a quantitative understanding of brain function, but it is difficult because of an upward bias due to limited experimental sampling. We present new progress, based on two basic insights, on reducing the bias problem. First, we show that by means of a careful application of data-shuffling techniques, it is possible to cancel almost entirely the bias of the noise entropy, the most biased part of information. This procedure provides a new information estimator that is much less biased than the standard direct one and has similar variance. Second, we use a nonparametric test to determine whether all the information encoded by the spike train can be decoded assuming a low-dimensional response model. If this is the case, the complexity of response space can be fully captured by a small number of easily sampled parameters. Combining these two different procedures, we obtain a new class of precise estimators of information quantities, which can provide data-robust upper and lower bounds to the mutual information. These bounds are tight even when the number of trials per stimulus available is one order of magnitude smaller than the number of possible responses. The effectiveness and the usefulness of the methods are tested through applications to simulated data and recordings from somatosensory cortex. This application shows that even in the presence of strong correlations, our methods constrain precisely the amount of information encoded by real spike trains recorded in vivo.

We study the relationship between the accuracy of a large neuronal population in encoding periodic sensory stimuli and the width of the tuning curves of individual neurons in the population. By using general simple models of population activity, we show that when considering one or two periodic stimulus features, a narrow tuning width provides better population encoding accuracy. When encoding more than two periodic stimulus features, the information conveyed by the population is instead maximal for finite values of the tuning width. These optimal values are only weakly dependent on model parameters and are similar to the width of tuning to orientation or motion direction of real visual cortical neurons. A very large tuning width leads to poor encoding accuracy, whatever the number of stimulus features encoded. Thus, optimal coding of periodic stimuli is different from that of nonperiodic stimuli, which, as shown in previous studies, would require infinitely large tuning widths when coding more than two stimulus features.

We develop new data-robust lower-bound methods to quantify the information carried by the timing of spikes emitted by neuronal populations. These methods have better sampling properties and are tighter than previous bounds based on neglecting correlation in the noise entropy. Our new lower bounds are precise also in the presence of strongly correlated firing. They are not precise only if correlations are strongly stimulus modulated over a long time range. Under conditions typical of many neurophysiological experiments, these techniques permit precise information estimates to be made even with data samples that are three orders of magnitude smaller than the size of the response space.

A typical neuron in visual cortex receives most inputs from other cortical neurons with a roughly similar stimulus preference. Does this arrange- ment of inputs allow efficient readout of sensory information by the tar- get cortical neuron? We address this issue by using simple modelling of neuronal population activity and information theoretic tools. We find that efficient synaptic information transmission requires that the tuning curve of the afferent neurons is approximately as wide as the spread of stim- ulus preferences of the afferent neurons reaching the target neuron. By meta analysis of neurophysiological data we found that this is the case for cortico-cortical inputs to neurons in visual cortex. We suggest that the organization of V1 cortico-cortical synaptic inputs allows optimal in- formation transmission. 1 Introduction A typical neuron in visual cortex receives most of its inputs from other visual cortical neu- rons. The majority of cortico-cortical inputs arise from afferent cortical neurons with a preference to stimuli which is similar to that of the target neuron [1, 2, 3]. For exam- ple, orientation selective neurons in superficial layers in ferret visual cortex receive more than 50% of their cortico-cortical excitatory inputs from neurons with orientation prefer- ence which is less than 30o apart. However, this input structure is rather broad in terms of stimulus-specificity: cortico-cortical connections between neurons tuned to dissimilar stimulus orientation also exist [4]. The structure and spread of the stimulus specificity of cortico-cortical connections has received a lot of attention because of its importance for understanding the mechanisms of generation of orientation tuning (see [4] for a review). However, little is still known on whether this structure of inputs allows efficient transmis- sion of sensory information across cortico-cortical synapses. It is likely that efficiency of information transmission across cortico-cortical synapses also depends on the width of tuning curves of the afferent cortical neurons to stimuli. In fact, theoretical work on population coding has shown that the width of the tuning curves has Corresponding author an important influence on the quality and the nature of the information encoding in cortical populations [5, 6, 7, 8]. Another factor that may influence the efficiency of cortico-cortical synaptic information transmission is the biophysical capability of the target neuron. To conserve all information during synaptic transmission, the target neuron must conserve the `label' of the spikes arriving from multiple input neurons at different places on its dendritic tree [9]. Because of biophysical limitations, a target neuron that e.g. can only sum inputs at the soma may lose a large part of the information present in the afferent activity. The optimal arrangement of cortico-cortical synapses may also depend on the capability of postsynaptic neurons in processing separately spikes from different neurons. In this paper, we address the problem of whether cortico-cortical synaptic systems encode information efficiently. We introduce a simple model of neuronal information processing that takes into account both the selective distribution of stimulus preferences typical of cortico-cortical connections and the potential biophysical limitations of cortical neurons. We use this model and information theoretic tools to investigate whether there is an opti- mal trade-off between the spread of distribution of stimulus preference across the afferent neurons and the tuning width of the afferent neurons itself. We find that efficient synaptic information transmission requires that the tuning curve of the afferent neurons is approx- imately as wide as the spread of stimulus preferences of the afferent fibres reaching the target neuron. By reviewing anatomical and physiological data, we argue that this optimal trade-off is approximately reached in visual cortex. These results suggest that neurons in visual cortex are wired to decode optimally information from a stimulus-specific distribu- tion of synaptic inputs. 2 Model of the activity of the afferent neuronal population We consider a simple model for the activity of the afferent neuronal population based on the known tuning properties and spatial and synaptic organisation of sensory areas. 2.1 Stimulus tuning of individual afferent neurons We assume that the the population is made of a large number N of neurons (for a real cortical neuron, the number N of afferents is in the order of few thousands [10]). The response of each neuron rk(k = 1, , N) is quantified as the number of spikes fired in a salient post-stimulus time window of a length . Thus, the overall neuronal population response is represented as a spike count vector r = (r1, , rN ). We assume that the neurons are tuned to a small number D of relevant stimulus parameters [11, 12], such as e.g. orientation, speed or direction of motion of a visual object. The stimulus variable will thus be described as a vector s = (s1, . . . , sD) of dimension D. The number of stimulus features that are encoded by the neuron will be left as a free parameter to be varied within the range 1-5 for two reasons. First, although there is evidence that the number of stimulus features encoded by a single neuron is limited [11, 12], more research is still needed to determine exactly how many stimulus parameters are encoded in different areas. Second, a previous related study [8] has shown that, when considering large neuronal populations with a uniform distribution of stimulus preferences (such as an hypercolumn in V1 containing all stimulus orientations) the tuning width of individual neurons which is optimal for population coding depends crucially on the number of stimulus features being encoded. Thus, it is interesting to investigate how the optimal arrangement of cortico- cortical synaptic systems depends on the number of stimulus features being encoded. The neuronal tuning function of the k - th neuron (k = 1, , N ), which quantifies the mean spike count of the k - th neuron to the presented stimulus, is modelled as a Gaussian distribution, characterised by the following parameters: preferred stimulus s(k), tuning width f , and response modulation m: - (s-s(k))2 f (k)(s) = me 2f 2 (1) The Gaussian tuning curve is a good description of the tuning properties of e.g. V1 or MT neurons to variables such as stimulus orientation motion direction [13, 14, 15], and is hence widely used in models of sensory coding [16, 17]. Large values of f indicate coarse coding, whereas small values of f indicate sharp tuning. Spike count responses of each neuron on each trial are assumed to follow a Poisson distri- bution whose mean is given by the above neuronal tuning function (Eq. 1). The Poisson model is widely used because it is the simplest model of neuronal firing that captures the salient property of neuronal firing that the variance of spike counts is proportional to its mean. The Poisson model neglects all correlations between spikes. This assumption is certainly a simplification but it is sufficient to account for the majority of the information transmitted by real cortical neurons [18, 19, 20] and, as we shall see later, it is mathemati- cally convenient because it makes our model tractable. 2.2 Distribution of stimulus preferences among the afferent population Neurons in sensory cortex receive a large number of inputs from other neurons with a vari- ety of stimulus preferences. However, the majority of their inputs come from neurons with roughly similar stimulus preference [1, 2, 3]. To characterise correctly this type of spread of stimulus preference among the afferent population, we assume (unlike in previous stud- ies [8]), that the probability distribution of the preferred stimulus among afferent neurons follows a Gaussian distribution: 1 - (^s-^s0)2 P (^s) = 22 p (2 e (2) )D/2D p In Eq. (2) the parameter ^ s0 represents the the center of the distribution, thus being the most represented preferred stimulus in the population. (we set, without loss of general- ity, ^ s0 = 0.) The parameter p controls the spread of stimulus preferences of the afferent neuronal population: a small value of p indicates that a large fraction of the population have similar stimulus preferences, and a large value of p indicates that all stimuli are represented similarly. A Gaussian distribution of stimulus preferences of the afferent pop- ulation fits well empirical data on distribution of preferred orientations of synaptic inputs of neurons in both deep and superficial layers of ferret primary visual cortex [3]. 3 Width of tuning and spread of stimulus preferences in visual cortex To estimate the width of tuning f and the spread of stimulus preferences p of cortico- cortical afferent populations in visual cortex, we reviewed critically published anatomical and physiological data. We concentrated on excitatory synaptic inputs, which form the majority of inputs to a cortical pyramidal neuron [10]. We computed p by fitting (by a least square method) the published histograms of synaptic connections as function of stimulus preference of the input neuron to Gaussian distributions. Similarly, we determined f by fitting spike count histograms to Gaussian tuning curves. When considering a target neuron in ferret primary visual cortex and using orientation as the stimulus parameters, the spread of stimulus preferences p of its inputs is 20o for layer 5/6 neurons [3], and 16o [3] to 23o [21] for layer 2/3 neurons. The orientation tuning width f of the cortical inputs to the V1 target neuron is that of other V1 neurons that project to it. This f is 17o for Layer 4 neurons [22], and it is similar for neurons in deep and superficial layers [3]. When considering a target neuron in Layer 4 of cat visual cortex and orientation tuning, the spread of stimulus preference p is 20o [2] and f is 17o. When considering a target neuron in ferret visual cortex and motion direction tuning, the spread of tuning of its inputs p is 30 o [1]. Motion direction tuning widths of macaque neurons is 28o, and this width is similar across species (see [13]). The most notable finding of our meta-analysis of published data is that p and f appear to be approximately of the same size and their ratio f /p is distributed around 1, in the range 0.7 to 1.1 for the above data. We will use our model to understand whether this range of f /p corresponds to an optimal way to transmit information across a synaptic system. 4 Information theoretic quantification of population decoding To characterise how a target neuronal system can decode the information about sensory stimuli contained in the activity of its afferent neuronal population, we use mutual infor- mation [23]. The mutual information between a set of stimuli and the neuronal responses quantifies how well any decoder can discriminate among stimuli by observing the neuronal responses. This measure has the advantage of being independent of the decoding mecha- nism used, and thus puts precise constraints on the information that can be decoded by any biological system operating on the afferent activity. Previous studies on the information content of an afferent neuronal population [7, 8] have assumed that the target neuronal decoding system can extract all the information during synaptic transmission. To do so, the target neuron must conserve the "label" of the spikes arriving from multiple neurons at different sites on its dendritic tree [9]. Given the poten- tial biophysical difficulty in processing each spike separately, a simple alternative to spike labelling has been proposed, - spike pooling [10, 24]. In this scheme, the target neuron simply sums up the afferent activity. To characterize how the decoding of afferent informa- tion would work in both cases, we compute both the information that can be decoded by either a system that processes separately spikes from different neurons (the "labeled-line information") and the information available to a decoder that sums all incoming spikes (the "pooled information") [9, 24]. In the next two subsections we define these quantities and we explain how we compute it in our model. 4.1 The information available to the the labeled-line decoder The mutual information between the set of the stimuli and the labeled-line neuronal popu- lation activity is defined as follows [9, 24]: ILL(S, R) = dsP (s) P (r|s) log P (r|s) r P (r) (3) where P (s) is the probability of stimulus occurrence (here taken for simplicity as a uni- form distribution over the hypersphere of D dimensions and `radius' s). P (r|s) is the probability of observing a neuronal population response r conditional to the occurrence of stimulus s, and P (r) = dsP (s)P (r|s). Since the response vector r keeps separate the spike counts of each neuron, the amount of information in Eq. (3) is only accessible to a decoder than can keep the label of which neuron fired which spike [9, 24]. The probability P (r|s) is computed according to the Poisson distribution, which is entirely determined by the knowledge of the tuning curves [5]. The labeled-line mutual information is difficult to compute for large populations, because it requires the knowledge of the probability of the large-dimensional response vector r. However, since in our model we assume that we have a very large number of independent neurons in the population and that the total activity of the system is of the order of its size, then we can use the following simpler (but still exact) expression[16, 25]: 1 ILL(S, R) = H(S) - D ln (2 2 e) + 2 ds P(s) ln (|J (s)|) (4) where H(S) is the entropy of the prior stimulus presentation distribution P (S), J (s) is the Fisher information matrix and | . . . | stands for the determinant. The Fisher information matrix is a D D matrix who's elements i, j are defined as follows: Ji,j(s) = - P (r|s) 2 log P(r|s) , (5) r si sj Fisher information is a useful measure of the accuracy with which a particular stimulus can be reconstructed from a single trial observation of neuronal population activity. However, in this paper it is used only as a step to obtain a computationally tractable expression for the labeled-line mutual information. The Fisher information matrix can be computed by taking into account that for a population of Poisson neurons is just the sum of the Fisher informa- tion for individual neurons, and the latter has a simple expression in terms of tuning curves [16]. Since the neuronal population size N is is large, the sum over Fisher information of individual neurons can be replaced by an integral over the stimulus preferences of the neurons in the population, weighted by their probability density P (^s). After performing the integral over the distribution of preferred stimuli, we arrived at the following result for the elements of the Fisher information matrix: J D-2 - 2 i,j(s) = N m i,j + 2 (i,j + ij) e 2(1+2) (6) 2p (1 + 2)D2 +2 where we have introduced the following short-hand notation f /p and s/p ; i,j stands for the Kroneker Delta. From Eq. (6) it is possible to compute explicitly the determinant |J (s)|, which has the following form: D |J (s)| = i = ()D(1 + 2)D-1 1 + 2(1 + 2) (7) i=1 where () is given by: D-2 - 2 () = N m e 2(1+2) (8) 2p (1 + 2)D2 +1 Inserting Eq. (7) into Eq. (4), one obtains a tractable but still exact expression for the mutual information , which has the advantage over Eq. (3) of requiring only an integral over a D-dimensional stimulus rather than a sum over an infinite population. We have studied numerically the dependence of the labeled-line information on the pa- rameters f and p as a function of the number of encoded stimulus features D 1. We investigated this by fixing p and then varying the ration f /p over a wide range. Results (obtained for p = 1 but representative of a wide f range) are reported in Fig. 1. We found that, unlike the case of a uniform distribution of stimulus preferences [8], there is a finite value of the width of tuning f that maximizes the information for all D 2. Inter- estingly, for D 2 the range 0.7 f /p 1.1 found in visual cortex either contains the maximum or corresponds to near optimal values of information transmission. For D = 1, information is maximal for very narrow tuning curves. However, also in this case the in- formation values are still efficient in the cortical range f /p 1, in that the tail of the D = 1 information curve is avoided in that region. Thus, the range of values of f and p found in visual cortex allows efficient synaptic information transmission over a wide range of number of stimulus features encoded by the neuron. 1We found (data not shown) that other parameters such as m and , had a weak or null effect on the optimal configuration; see [17] for a D = 1 example in a different context. D=1 (S,R) LL I D=5 0 2 4 6 8 / f p Figure 1: Mutual labeled-line information as a function of the ratio of tuning curve width and stimulus preference spread f /p. The curves for each stimulus dimensionality D were shifted by a constant factor to separate them for visual inspection (lower curves cor- respond to higher values of D). The y-axis is thus in arbitrary units. The position of the maximal information for each stimulus dimension falls either inside the range of values of f /p found in visual cortex, or very close to it (see text) . Parameters are as follows: s = 2, rmax = 50Hz, = 10ms. 4.2 The information available to the the pooling decoder We now consider the case in which the target neuron cannot process separately spikes from different neurons (for example, a neuron that just sums up post-synaptic potentials of approximately equal weight at the soma). In this case the label of the neuron that fired each spike is lost by the target neuron, and it can only operate on the pooled neuronal signal, in which the identity of each spike is lost. Pooling mechanisms have been proposed as simple information processing strategies for the nervous system. We now study how pooling changes the requirements for efficient decoding by the target neuron. Mathematically speaking, pooling maps the vector r onto a scalar equal to the sum of the individual activities: = rk. Thus, the mutual information that can be extracted by any decoder that only pools it inputs is given by the following expression: IP (S, R) = dsP (s) P (|s) log P (|s) P () (9) where P (|s) and P () are the the stimulus-conditional and stimulus-unconditional proba- bility of observing a pooled population response on a single trial. The probability P (|s) can be computed by noting that a sum of Poisson-distributed responses is still a Poisson- distributed response whose tuning curve to stimuli is just the sum of the individual tuning curves. The pooled mutual information is thus a function of a single Poisson-distributed response variables and can be computed easily also for large populations. The dependence of the pooled information on the parameters f and p as a function of the number of encoded stimulus features D is reported in Fig. 2. There is one important difference with respect to the labeled-line information transmission case. The difference is that, for the pooled information, there is a finite optimal value for information transmission also when the neurons are tuned to one-dimensional stimulus feature. For all cases of stim- ulus dimensionality considered, the efficient information transmission though the pooled D=1 (S,R) D=3 P I 0 1 2 3 4 / f p Figure 2: Pooled mutual information as a function of the ratio of tuning curve width and stimulus preference spread f /p. The maxima are inside the range of experimental values of f /p found in the visual cortex, or very close to it (see text). As for Fig. 1, the curves for each stimulus dimensionality D were shifted by a constant factor to separate them for visual inspection (lower curves correspond to higher values of D). The y-axis is thus in arbitrary units. Parameters are as follows: s = 2, rmax = 50 Hz, = 10ms. neuronal decoder can still be reached in the visual cortical range 0.7 f p 1.1. This finding shows that the range of values of f and p found in visual cortex allows effi- cient synaptic information transmission even if the target neuron does not rely on complex dendritic processing to conserve the label of the neuron that fired the spike.

We demonstrate that the information contained in the spike occurrence times of a population of neurons can be broken up into a series of terms, each reflecting something about potential coding mechanisms. This is possible in the coding regime in which few spikes are emitted in the relevant time window. This approach allows us to study the additional information contributed by spike timing beyond that present in the spike counts and to examine the contributions to the whole information of different statistical properties of spike trains, such as firing rates and correlation functions. It thus forms the basis for a new quantitative procedure for analyzing simultaneous multiple neuron recordings and provides theoretical constraints on neural coding strategies. We find a transition between two coding regimes, depending on the size of the relevant observation timescale. For time windows shorter than the timescale of the stimulus-induced response fluctuations, there exists a spike count coding phase, in which the purely temporal information is of third order in time. For time windows much longer than the characteristic timescale, there can be additional timing information of first order, leading to a temporal coding phase in which timing information may affect the instantaneous information rate. In this new framework, we study the relative contributions of the dynamic firing rate and correlation variables to the full temporal information, the interaction of signal and noise correlations in temporal coding, synergy between spikes and between cells, and the effect of refractoriness. We illustrate the utility of the technique by analyzing a few cells from the rat barrel cortex.

The effectiveness of various stimulus identification (decoding) procedures for extracting the information carried by the responses of a population of neurons to a set of repeatedly presented stimuli is studied analytically, in the limit of short time windows. It is shown that in this limit, the entire information content of the responses can sometimes be decoded, and when this is not the case, the lost information is quantified. In particular, the mutual information extracted by taking into account only the most likely stimulus in each trial turns out to be, if not equal, much closer to the true value than that calculated from all the probabilities that each of the possible stimuli in the set was the actual one. The relation between the mutual information extracted by decoding and the percentage of correct stimulus decodings is also derived analytically in the same limit, showing that the metric content index can be estimated reliably from a few cells recorded from brief periods. Computer simulations as well as the activity of real neurons recorded in the primate hippocampus serve to confirm these results and illustrate the utility and limitations of the approach.

The distribution of responses of sensory neurons to ecological stimulation has been proposed to be designed to maximize information transmission, which according to a simple model would imply an exponential distribution of spike counts in a given time window. We have used recordings from inferior temporal cortex neurons responding to quasi-natural visual stimulation (presented using a video of everyday lab scenes and a large number of static images of faces and natural scenes) to assess the validity of this exponential model and to develop an alternative simple model of spike count distributions. We find that the exponential model has to be rejected in 84% of cases (at the p < 0.01 level). A new model, which accounts for the firing rate distribution found in terms of slow and fast variability in the inputs that produce neuronal activation, is rejected statistically in only 16% of cases. Finally, we show that the neurons are moderately efficient at transmitting information but not optimally efficient.

It is difficult to extract the information carried by neuronal responses about a set of stimuli because limited data samples result in biased estimates. Recently two improved procedures have been developed to calculate information from experimental results: a binning-and-correcting procedure and a neural network procedure. We have used data produced from a model of the spatiotemporal receptive fields of parvocellular and magnocellular lateral geniculate neurons to study the performance of these methods as a function of the number of trials used. Both procedures yield accurate results for one-dimensional neuronal codes. They can also be used to produce a reasonable estimate of the extra information in a three-dimensional code, in this instance, within 0.05-0.1 bit of the asymptotically calculated value--about 10% of the total transmitted information. We believe that this performance is much more accurate than previous procedures.

Extracting information measures from limited experimental samples, such as those normally available when using data recorded in vivo from mammalian cortical neurons, is known to be plagued by a systematic error, which tends to bias the estimate upward. We calculate here the average of the bias, under certain conditions, as an asymptotic expansion in the inverse of the size of the data sample. The result agrees with numerical simulations, and is applicable, as an additive correction term, to measurements obtained under such conditions. Moreover, we discuss the implications for measurements obtained through other usual procedures.