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This letter proposes a novel method, multi-input, multi-output neuronal mode network (MIMO-NMN), for modeling encoding dynamics and functional connectivity in neural ensembles such as the hippocampus. Compared with conventional approaches such as the Volterra-Wiener model, linear-nonlinear-cascade (LNC) model, and generalized linear model (GLM), the NMN has several advantages in terms of estimation accuracy, model interpretation, and functional connectivity analysis. We point out the limitations of current neural spike modeling methods, especially the estimation biases caused by the imbalanced class problem when the number of zeros is significantly larger than ones in the spike data. We use synthetic data to test the performance of NMN with a comparison of the traditional methods, and the results indicate the NMN approach could reduce the imbalanced class problem and achieve better predictions. Subsequently, we apply the MIMO-NMN method to analyze data from the human hippocampus. The results indicate that the MIMO-NMN method is a promising approach to modeling neural dynamics and analyzing functional connectivity of multi-neuronal data.

Neurostimulation is a promising therapy for abating epileptic seizures. However, it is extremely difficult to identify optimal stimulation patterns experimentally. In this study, human recordings are used to develop a functional 24 neuron network statistical model of hippocampal connectivity and dynamics. Spontaneous seizure-like activity is induced in silico in this reconstructed neuronal network. The network is then used as a testbed to design and validate a wide range of neurostimulation patterns. Commonly used periodic trains were not able to permanently abate seizures at any frequency. A simulated annealing global optimization algorithm was then used to identify an optimal stimulation pattern, which successfully abated 92% of seizures. Finally, in a fully responsive, or closed-loop, neurostimulation paradigm, the optimal stimulation successfully prevented the network from entering the seizure state. We propose that the framework presented here for algorithmically identifying patient-specific neurostimulation patterns can greatly increase the efficacy of neurostimulation devices for seizures.

This letter examines the results of input-output (nonparametric) modeling based on the analysis of data generated by a mechanism-based (parametric) model of CA3-CA1 neuronal connections in the hippocampus. The motivation is to obtain biological insight into the interpretation of such input-output (Volterra-equivalent) models estimated from synthetic data. The insights obtained may be subsequently used to interpretat input-output models extracted from actual experimental data. Specifically, we found that a simplified parametric model may serve as a useful tool to study the signal transformations in the hippocampal CA3-CA1 regions. Input-output modeling of model-based synthetic data show that GABAergic interneurons are responsible for regulating neuronal excitation, controlling the precision of spike timing, and maintaining network oscillations, in a manner consistent with previous studies. The input-output model obtained from real data exhibits intriguing similarities with its synthetic-data counterpart, demonstrating the importance of a dynamic resonance in the system/model response around 2 Hz to 3 Hz. Using the input-output model from real data as a guide, we may be able to amend the parametric model by incorporating more mechanisms in order to yield better-matching input-output model. The approach we present can also be applied to the study of other neural systems and pathways.

In this paper, we have introduced a general modeling approach for dynamic nonlinear systems that utilizes a variant of the simulated annealing algorithm for training the Laguerre-Volterra network (LVN) to overcome the local minima and convergence problems and employs a pruning technique to achieve sparse LVN representations with l 1 regularization. We tested this new approach with computer simulated systems and extended it to autoregressive sparse LVN (ASLVN) model structures that are suitable for input-output modeling of nonlinear systems that exhibit transitions in dynamic states, such as the Hodgkin-Huxley (H-H) equations of neuronal firing. Application of the proposed ASLVN to the H-H equations yields a more parsimonious input-output model with improved predictive capability that is amenable to more insightful physiological/biological interpretation.

Very productive collaborative investigations characterized how multineuron hippocampal ensembles recorded in nonhuman primates (NHPs) encode short-term memory necessary for successful performance in a delayed match to sample (DMS) task and utilized that information to devise a unique nonlinear multi-input multi-output (MIMO) memory prosthesis device to enhance short-term memory in real-time during task performance. Investigations have characterized how the hippocampus in primate brain encodes information in a multi-item, rule-controlled, delayed match to sample (DMS) task. The MIMO model was applied via closed loop feedback micro-current stimulation during the task via conformal electrode arrays and enhanced performance of the complex memory requirements. These findings clearly indicate detection of a means by which the hippocampus encodes information and transmits this information to other brain regions involved in memory processing. By employing the nonlinear dynamic multi-input/multi-output (MIMO) model, developed and adapted to hippocampal neural ensemble firing patterns derived from simultaneous recorded multi-neuron CA1 and CA3 activity, it was possible to extract information encoded in the Sample phase of DMS trials that was necessary for successful performance in the subsequent Match phase of the task. The extension of this MIMO model to online delivery of electrical stimulation patterns to the same recording loci that exhibited successful CA1 firing in the DMS Sample Phase provided the means to increase task performance on a trial-by-trial basis. Increased utility of the MIMO model as a memory prosthesis was exhibited by the demonstration of cumulative increases in DMS task performance with repeated MIMO stimulation over many sessions. These results, reported below in this article, provide the necessary demonstrations to further the feasibility of the MIMO model as a memory prosthesis to recover and/or enhance encoding of cognitive information in humans with memory disruptions resulting from brain injury, disease or aging., (Copyright © 2016 Elsevier Inc. All rights reserved.)

To understand how memory information is encoded in the hippocampus, we build classification models to decode memory features from hippocampal CA3 and CA1 spatio-temporal patterns of spikes recorded from epilepsy patients performing a memory-dependent delayed match-to-sample task. The classification model consists of a set of B-spline basis functions for extracting memory features from the spike patterns, and a sparse logistic regression classifier for generating binary categorical output of memory features. Results show that classification models can extract significant amount of memory information with respects to types of memory tasks and categories of sample images used in the task, despite the high level of variability in prediction accuracy due to the small sample size. These results support the hypothesis that memories are encoded in the hippocampal activities and have important implication to the development of hippocampal memory prostheses.

Objective: Traditional hippocampal modeling has focused on the series of feedforward synapses known as the trisynaptic pathway. However, feedback connections from CA1 back to the hippocampus through the entorhinal cortex (EC) actually make the hippocampus a closed-loop system. By constructing a functional closed-loop model of the hippocampus, one may learn how both physiological and epileptic oscillations emerge and design efficient neurostimulation patterns to abate such oscillations., Approach: Point process input-output models where estimated from recorded rodent hippocampal data to describe the nonlinear dynamical transformation from CA3 → CA1, via the schaffer-collateral synapse, and CA1 → CA3 via the EC. Each Volterra-like subsystem was composed of linear dynamics (principal dynamic modes) followed by static nonlinearities. The two subsystems were then wired together to produce the full closed-loop model of the hippocampus., Main Results: Closed-loop connectivity was found to be necessary for the emergence of theta resonances as seen in recorded data, thus validating the model. The model was then used to identify frequency parameters for the design of neurostimulation patterns to abate seizures., Significance: Deep-brain stimulation (DBS) is a new and promising therapy for intractable seizures. Currently, there is no efficient way to determine optimal frequency parameters for DBS, or even whether periodic or broadband stimuli are optimal. Data-based computational models have the potential to be used as a testbed for designing optimal DBS patterns for individual patients. However, in order for these models to be successful they must incorporate the complex closed-loop structure of the seizure focus. This study serves as a proof-of-concept of using such models to design efficient personalized DBS patterns for epilepsy.

This paper presents a systems identification approach for studying the long-term synaptic plasticity using natural spiking activities. This approach consists of three modeling steps. First, a multi-input, single-output (MISO), nonlinear dynamical spiking neuron model is formulated to estimate and represent the synaptic strength in means of functional connectivity between input and output neurons. Second, this MISO model is extended to a nonstationary form to track the time-varying properties of the synaptic strength. Finally, a Volterra modeling method is used to extract the synaptic learning rule, e.g., spike-timing-dependent plasticity, for the explanation of the input-output nonstationarity as the consequence of the past input-output spiking patterns. This framework is developed to study the underlying mechanisms of learning and memory formation in behaving animals, and may serve as the computational basis for building the next-generation adaptive cortical prostheses., (Copyright © 2014 Elsevier B.V. All rights reserved.)

We develop an autoregressive model framework based on the concept of Principal Dynamic Modes (PDMs) for the process of action potential (AP) generation in the excitable neuronal membrane described by the Hodgkin-Huxley (H-H) equations. The model's exogenous input is injected current, and whenever the membrane potential output exceeds a specified threshold, it is fed back as a second input. The PDMs are estimated from the previously developed Nonlinear Autoregressive Volterra (NARV) model, and represent an efficient functional basis for Volterra kernel expansion. The PDM-based model admits a modular representation, consisting of the forward and feedback PDM bases as linear filterbanks for the exogenous and autoregressive inputs, respectively, whose outputs are then fed to a static nonlinearity composed of polynomials operating on the PDM outputs and cross-terms of pair-products of PDM outputs. A two-step procedure for model reduction is performed: first, influential subsets of the forward and feedback PDM bases are identified and selected as the reduced PDM bases. Second, the terms of the static nonlinearity are pruned. The first step reduces model complexity from a total of 65 coefficients to 27, while the second further reduces the model coefficients to only eight. It is demonstrated that the performance cost of model reduction in terms of out-of-sample prediction accuracy is minimal. Unlike the full model, the eight coefficient pruned model can be easily visualized to reveal the essential system components, and thus the data-derived PDM model can yield insight into the underlying system structure and function.

Although an anatomical connection from CA1 to CA3 via the Entorhinal Cortex (EC) and through backprojecting interneurons has long been known it exist, it has never been examined quantitatively on the single neuron level, in the in-vivo nonpatholgical, nonperturbed brain. Here, single spike activity was recorded using a multi-electrode array from the CA3 and CA1 areas of the rodent hippocampus (N = 7) during a behavioral task. The predictive power from CA3→CA1 and CA1→CA3 was examined by constructing Multivariate Autoregressive (MVAR) models from recorded neurons in both directions. All nonsignificant inputs and models were identified and removed by means of Monte Carlo simulation methods. It was found that 121/166 (73 %) CA3→CA1 models and 96/145 (66 %) CA1→CA3 models had significant predictive power, thus confirming a predictive 'Granger' causal relationship from CA1 to CA3. This relationship is thought to be caused by a combination of truly causal connections such as the CA1→EC→CA3 pathway and common inputs such as those from the Septum. All MVAR models were then examined in the frequency domain and it was found that CA3 kernels had significantly more power in the theta and beta range than those of CA1, confirming CA3's role as an endogenous hippocampal pacemaker.

In order to build hippocampal prostheses for restoring memory functions, we build multi-input, multi-output (MIMO) nonlinear dynamical models of the human hippocampus. Spike trains are recorded from the hippocampal CA3 and CA1 regions of epileptic patients performing a memory-dependent delayed match-to-sample task. Using CA3 and CA1 spike trains as inputs and outputs respectively, second-order sparse generalized Laguerre-Volterra models are estimated with group lasso and local coordinate descent methods to capture the nonlinear dynamics underlying the spike train transformations. These models can accurately predict the CA1 spike trains based on the ongoing CA3 spike trains and thus will serve as the computational basis of the hippocampal memory prosthesis.

A methodology for nonlinear modeling of multi-input multi-output (MIMO) neuronal systems is presented that utilizes the concept of Principal Dynamic Modes (PDM). The efficacy of this new methodology is demonstrated in the study of the dynamic interactions between neuronal ensembles in the Pre-Frontal Cortex (PFC) of a behaving non-human primate (NHP) performing a Delayed Match-to-Sample task. Recorded spike trains from Layer-2 and Layer-5 neurons were viewed as the "inputs" and "outputs", respectively, of a putative MIMO system/model that quantifies the dynamic transformation of multi-unit neuronal activity between Layer-2 and Layer-5 of the PFC. Model prediction performance was evaluated by means of computed Receiver Operating Characteristic (ROC) curves. The PDM-based approach seeks to reduce the complexity of MIMO models of neuronal ensembles in order to enable the practicable modeling of large-scale neural systems incorporating hundreds or thousands of neurons, which is emerging as a preeminent issue in the study of neural function. The "scaling-up" issue has attained critical importance as multi-electrode recordings are increasingly used to probe neural systems and advance our understanding of integrated neural function. The initial results indicate that the PDM-based modeling methodology may greatly reduce the complexity of the MIMO model without significant degradation of performance. Furthermore, the PDM-based approach offers the prospect of improved biological/physiological interpretation of the obtained MIMO models.

We propose a new variant of Volterra-type model with a nonlinear auto-regressive (NAR) component that is a suitable framework for describing the process of AP generation by the neuron membrane potential, and we apply it to input-output data generated by the Hodgkin-Huxley (H-H) equations. Volterra models use a functional series expansion to describe the input-output relation for most nonlinear dynamic systems, and are applicable to a wide range of physiologic systems. It is difficult, however, to apply the Volterra methodology to the H-H model because is characterized by distinct subthreshold and suprathreshold dynamics. When threshold is crossed, an autonomous action potential (AP) is generated, the output becomes temporarily decoupled from the input, and the standard Volterra model fails. Therefore, in our framework, whenever membrane potential exceeds some threshold, it is taken as a second input to a dual-input Volterra model. This model correctly predicts membrane voltage deflection both within the subthreshold region and during APs. Moreover, the model naturally generates a post-AP afterpotential and refractory period. It is known that the H-H model converges to a limit cycle in response to a constant current injection. This behavior is correctly predicted by the proposed model, while the standard Volterra model is incapable of generating such limit cycle behavior. The inclusion of cross-kernels, which describe the nonlinear interactions between the exogenous and autoregressive inputs, is found to be absolutely necessary. The proposed model is general, non-parametric, and data-derived.

A major factor involved in providing closed loop feedback for control of neural function is to understand how neural ensembles encode online information critical to the final behavioral endpoint. This issue was directly assessed in rats performing a short-term delay memory task in which successful encoding of task information is dependent upon specific spatio-temporal firing patterns recorded from ensembles of CA3 and CA1 hippocampal neurons. Such patterns, extracted by a specially designed nonlinear multi-input multi-output (MIMO) nonlinear mathematical model, were used to predict successful performance online via a closed loop paradigm which regulated trial difficulty (time of retention) as a function of the "strength" of stimulus encoding. The significance of the MIMO model as a neural prosthesis has been demonstrated by substituting trains of electrical stimulation pulses to mimic these same ensemble firing patterns. This feature was used repeatedly to vary "normal" encoding as a means of understanding how neural ensembles can be "tuned" to mimic the inherent process of selecting codes of different strength and functional specificity. The capacity to enhance and tune hippocampal encoding via MIMO model detection and insertion of critical ensemble firing patterns shown here provides the basis for possible extension to other disrupted brain circuitry.

Intracortical recordings comprise both fast events, action potentials (APs), and slower events, known as local field potentials (LFPs). Although it is believed that LFPs mostly reflect local synaptic activity, it is unclear which of their signal components are most closely related to synaptic potentials and would therefore be causally related to the occurrence of individual APs. This issue is complicated by the significant contribution from AP waveforms, especially at higher LFP frequencies. In recordings of single-cell activity and LFPs from the human temporal cortex, we computed quantitative, nonlinear, causal dynamic models for the prediction of AP timing from LFPs, at millisecond resolution, before and after removing AP contributions to the LFP. In many cases, the timing of a significant number of single APs could be predicted from spike-free LFPs at different frequencies. Not surprisingly, model performance was superior when spikes were not removed. Cells whose activity was predicted by the spike-free LFP models generally fell into one of two groups: in the first group, neuronal spike activity was associated with specific phases of low LFP frequencies, lower spike activity at high LFP frequencies, and a stronger linear component in the spike-LFP model; in the second group, neuronal spike activity was associated with larger amplitude of high LFP frequencies, less frequent phase locking, and a stronger nonlinear model component. Spike timing in the first group was better predicted by the sign and level of the LFP preceding the spike, whereas spike timing in the second group was better predicted by LFP power during a certain time window before the spike.

The right level of abstraction for a model mimicking a neural function is often difficult to determine. There are trade-offs between capturing biological complexities on one hand and the scalability and efficiency of the model on the other. In this work, we describe a nonlinear Laguerre-Volterra model of the synaptic temporal integration of input spikes to postsynaptic potentials. This model is then efficiently implemented using analog subthreshold circuits and can serve as a foundation for future large-scale hardware systems that can emulate multi-input multi-output (MIMO) spike transformations in populations of neurons. The normalized mean square error in estimating real data using the circuit implementation of this model is less than 15%. The model components are modular and its parameters are adjustable for modeling temporal integration by neurons in other brain regions. The total power consumption of this nonlinear Laguerre-Volterra system is less than 5nW.

This paper presents a pilot application of the Boolean-Volterra modeling methodology presented in the companion paper (Part I) that is suitable for the analysis of systems with point-process inputs and outputs (e.g., recordings of the activity of neuronal ensembles). This application seeks to discover the causal links between two neuronal ensembles in the hippocampus of a behaving rat. The experimental data come from multi-unit recordings in the CA3 and CA1 regions of the hippocampus in the form of sequences of action potentials-treated mathematically as point-processes and computationally as spike-trains-that are collected in vivo during two behavioral tasks. The modeling objective is to identify and quantify the causal links among the neurons generating the recorded activity, using Boolean-Volterra models estimated directly from the data according to the methodological framework presented in the companion paper. The obtained models demonstrate the feasibility of the proposed approach using short data-records and provide some insights into the functional properties of the system (e.g., regarding the presence of rhythmic characteristics in the neuronal dynamics of these ensembles), making the proposed methodology an attractive tool for the analysis and modeling of multi-unit recordings from neuronal systems in a practical context.

This paper presents a new modeling approach for neural systems with point-process (spike) inputs and outputs that utilizes Boolean operators (i.e. modulo 2 multiplication and addition that correspond to the logical AND and OR operations respectively, as well as the AND_NOT logical operation representing inhibitory effects). The form of the employed mathematical models is akin to a "Boolean-Volterra" model that contains the product terms of all relevant input lags in a hierarchical order, where terms of order higher than first represent nonlinear interactions among the various lagged values of each input point-process or among lagged values of various inputs (if multiple inputs exist) as they reflect on the output. The coefficients of this Boolean-Volterra model are also binary variables that indicate the presence or absence of the respective term in each specific model/system. Simulations are used to explore the properties of such models and the feasibility of their accurate estimation from short data-records in the presence of noise (i.e. spurious spikes). The results demonstrate the feasibility of obtaining reliable estimates of such models, with excitatory and inhibitory terms, in the presence of considerable noise (spurious spikes) in the outputs and/or the inputs in a computationally efficient manner. A pilot application of this approach to an actual neural system is presented in the companion paper (Part II).

This paper presents a synergistic parametric and non-parametric modeling study of short-term plasticity (STP) in the Schaffer collateral to hippocampal CA1 pyramidal neuron (SC) synapse. Parametric models in the form of sets of differential and algebraic equations have been proposed on the basis of the current understanding of biological mechanisms active within the system. Non-parametric Poisson-Volterra models are obtained herein from broadband experimental input-output data. The non-parametric model is shown to provide better prediction of the experimental output than a parametric model with a single set of facilitation/depression (FD) process. The parametric model is then validated in terms of its input-output transformational properties using the non-parametric model since the latter constitutes a canonical and more complete representation of the synaptic nonlinear dynamics. Furthermore, discrepancies between the experimentally-derived non-parametric model and the equivalent non-parametric model of the parametric model suggest the presence of multiple FD processes in the SC synapses. Inclusion of an additional set of FD process in the parametric model makes it replicate better the characteristics of the experimentally-derived non-parametric model. This improved parametric model in turn provides the requisite biological interpretability that the non-parametric model lacks.

Parametric and non-parametric modeling methods are combined to study the short-term plasticity (STP) of synapses in the central nervous system (CNS). The nonlinear dynamics of STP are modeled by means: (1) previously proposed parametric models based on mechanistic hypotheses and/or specific dynamical processes, and (2) non-parametric models (in the form of Volterra kernels) that transforms the presynaptic signals into postsynaptic signals. In order to synergistically use the two approaches, we estimate the Volterra kernels of the parametric models of STP for four types of synapses using synthetic broadband input-output data. Results show that the non-parametric models accurately and efficiently replicate the input-output transformations of the parametric models. Volterra kernels provide a general and quantitative representation of the STP.

To understand the function of a brain region, e.g., hippocampus, it is necessary to model its input-output property. Such a model can serve as the computational basis of the development of cortical prostheses restoring the transformation of population neural activities performed by the brain region. We formulate a sparse generalized Laguerre-Volterra model (SGLVM) for the multiple-input, multiple-output (MIMO) transformation of spike trains. A SGLVM consists of a set of feedforward Laguerre-Volterra kernels, a feedback Laguerre-Volterra kernel, and a probit link function. The sparse model representation involving only significant self and cross terms is achieved through statistical model selection and cross-validation methods. The SGLVM is applied successfully to the hippocampal CA3-CA1 population dynamics.

The increasing availability of multiunit recordings gives new urgency to the need for effective analysis of "multidimensional" time-series data that are derived from the recorded activity of neuronal ensembles in the form of multiple sequences of action potentials--treated mathematically as point-processes and computationally as spike-trains. Whether in conditions of spontaneous activity or under conditions of external stimulation, the objective is the identification and quantification of possible causal links among the neurons generating the observed binary signals. A multiple-input/multiple-output (MIMO) modeling methodology is presented that can be used to quantify the neuronal dynamics of causal interrelationships in neuronal ensembles using spike-train data recorded from individual neurons. These causal interrelationships are modeled as transformations of spike-trains recorded from a set of neurons designated as the "inputs" into spike-trains recorded from another set of neurons designated as the "outputs." The MIMO model is composed of a set of multiinput/single-output (MISO) modules, one for each output. Each module is the cascade of a MISO Volterra model and a threshold operator generating the output spikes. The Laguerre expansion approach is used to estimate the Volterra kernels of each MISO module from the respective input-output data using the least-squares method. The predictive performance of the model is evaluated with the use of the receiver operating characteristic (ROC) curve, from which the optimum threshold is also selected. The Mann-Whitney statistic is used to select the significant inputs for each output by examining the statistical significance of improvements in the predictive accuracy of the model when the respective inputs is included. Illustrative examples are presented for a simulated system and for an actual application using multiunit data recordings from the hippocampus of a behaving rat.

The dentate gyrus is the first region of the hippocampus that receives and integrates sensory information (e.g., visual, auditory, and olfactory) via the perforant path, which is composed of two distinct neuronal pathways: the Lateral Perforant Path (LPP) and the Medial Perforant Path (MPP). This paper examines the nonlinear dynamic interactions among arbitrary stimulation patterns at these two afferent pathways and their combined effect on the resulting response of the granule cells at the dentate gyrus. We employ non-parametric Poisson-Volterra models that serve as canonical quantitative descriptors of the nonlinear dynamic transformations of the neuronal signals propagating through these two neuronal pathways. These Poisson-Volterra models are estimated in the so-called "reduced form" with experimental data from in vitro hippocampal slices and provide excellent predictions of the electrophysiological activity of the granule cells in response to arbitrary stimulation patterns. The data are acquired through a custom-made multi-electrode-array system, which stimulated simultaneously the two pathways with random impulse trains and recorded the neuronal postsynaptic activity at the granule cell layer. The results of this study show that significant nonlinear interactions exist between the LPP and the MPP that may be critical for the integration of sensory information performed by the dentate gyrus of the hippocampus.

A modeling approach to characterize the nonlinear dynamic transformations of the dentate gyrus of the hippocampus is presented and experimentally validated. The dentate gyrus is the first region of the hippocampus which receives and integrates sensory information via the perforant path. The perforant path is composed of two distinct pathways: 1) the lateral path and 2) the medial perforant path. The proposed approach examines and captures the short-term dynamic characteristics of these two pathways using a nonparametric, third-order Poisson-Volterra model. The nonlinear characteristics of the two pathways are represented by Poisson-Volterra kernels, which are quantitative descriptors of the nonlinear dynamic transformations. The kernels were computed with experimental data from in vitro hippocampal slices. The electrophysiological activity was measured with custom-made multielectrode arrays, which allowed selective stimulation with random impulse trains and simultaneous recordings of extracellular field potential activity. The results demonstrate that this mathematically rigorous approach is suitable for the multipathway complexity of the hippocampus and yields interpretable models that have excellent predictive capabilities. The resulting models not only accurately predict previously reported electrophysiological descriptors, such as paired pulses, but more important, can be used to predict the electrophysiological activity of dentate granule cells to arbitrary stimulation patterns at the perforant path.

Implementation of neuroprosthetic devices requires a reliable and accurate quantitative representation of the input-output transformations performed by the involved neuronal populations. Nonparametric, data driven models with predictive capabilities are excellent candidates for these purposes. When modeling input-output relations in multi-input neuronal systems, it is important to select the subset of inputs that are functionally and causally related to the output. Inputs that do not convey information about the actual transformation not only increase the computational burden but also affect the generalization of the model. Moreover, a reliable functional connectivity measure can provide patterns of information flow that can be linked to physiological and anatomical properties of the system. We propose a method based on the Volterra modeling approach that selects distinct subsets of inputs for each output based on the prediction of the respective models and its statistical evaluation. The algorithm builds successive models with increasing number of inputs and examines whether the inclusion of additional inputs benefits the predictive accuracy of the overall model. It also explores possible second-order (inter-modulatory) interactions among the inputs. The method was applied to multi-unit recordings from the CA3 (input) and CA1 (output) regions of the hippocampus in behaving rats, in order to reveal spatiotemporal connectivity maps of the input-output transformation taking place in the CA3-CA1 synapse.

One of the fundamental principles of cortical brain regions, including the hippocampus, is that information is represented in the ensemble firing of populations of neurons, i.e., spatio-temporal patterns of electrophysiological activity. The hippocampus has long been known to be responsible for the formation of declarative, or fact-based, memories. Damage to the hippocampus disrupts the propagation of spatio-temporal patterns of activity through hippocampal internal circuitry, resulting in a severe anterograde amnesia. Developing a neural prosthesis for the damaged hippocampus requires restoring this multiple-input, multiple-output transformation of spatio-temporal patterns of activity. Because the mechanisms underlying synaptic transmission and generation of electrical activity in neurons are inherently nonlinear, any such prosthesis must be based on a nonlinear multiple-input, multiple-output model. In this paper, we have formulated the transformational process of multi-site propagation of spike activity between two subregions of the hippocampus (CA3 and CA1) as the identification of a multiple-input, multiple-output (MIMO) system, and proposed that it can be decomposed into a series of multiple-input, single-output (MISO) systems. Each MISO system is modeled as a physiologically plausible structure that consists of 1) linear/nonlinear feedforward Volterra kernels modeling synaptic transmission and dendritic integration, 2) a linear feedback Volterra kernel modeling spike-triggered after-potentials, 3) a threshold for spike generation, 4) a summation process for somatic integration, and 5) a noise term representing intrinsic neuronal noise and the contributions of unobserved inputs. Input and output spike trains were recorded from hippocampal CA3 and CA1 regions of rats performing a spatial delayed-nonmatch-to-sample memory task that requires normal hippocampal function. Kernels were expanded with Laguerre basis functions and estimated using a maximum-likelihood method. Complexity of the feedforward kernel was progressively increased to capture higher-order system nonlinear dynamics. Results showed higher prediction accuracies as kernel complexity increased. Self-kernels describe the nonlinearities within each input. Cross-kernels capture the nonlinear interaction between inputs. Second- and third-order nonlinear models were found to successfully predict the CA1 output spike distribution based on CA3 input spike trains. First-order, linear models were shown to be insufficient.

The encoding of mechanical stimuli into action potentials in two types of spider mechanoreceptor neurons is modeled by use of the principal dynamic modes (PDM) methodology. The PDM model is equivalent to the general Wiener-Bose model and consists of a minimum set of linear dynamic filters (PDMs), followed by a multivariate static nonlinearity and a threshold function. The PDMs are obtained by performing eigen-decomposition of a matrix constructed using the first-order and second-order Volterra kernels of the system, which are estimated by means of the Laguerre expansion technique, utilizing measurements of pseudorandom mechanical stimulation (input signal) and the resulting action potentials (output signal). The static nonlinearity, which can be viewed as a measure of the probability of action potential firing as a function of the PDM output values, is computed as the locus of points of the latter that correspond to output action potentials. The performance of the model is assessed by computing receiver operating characteristic (ROC) curves, akin to the ones used in decision theory and quantified by computing the area under the ROC curve. Three PDMs are revealed by the analysis. The first PDM exhibits a high-pass characteristic, illustrating the importance of the velocity of slit displacement in the generation of action potentials at the mechanoreceptor output, while the second and third PDMs exhibit band-pass and low-pass characteristics, respectively. The corresponding three-input nonlinearity exhibits asymmetric behavior with respect to its arguments, suggesting directional dependence of the mechanoreceptor response on the mechanical stimulation and the PDM outputs, in agreement to our findings from a previous study (Ann Biomed Eng 27:391-402, 1999). Differences between the Type A and B neurons are observed in the zeroth-order Volterra kernels (related to the average firing), as well as in the magnitudes of the second and third PDMs that perform band-pass and low-pass processing of the input signal, respectively.

A new modeling approach of hippocampal nonlinear dynamics is presented. It is based on Principal Dynamics Modes (PDMs) derived from the Volterra kernels quantifying the hippocampal transformations. The approach is illustrated using data obtained from acute hippocampal slice preparations and from behaving rats performs a memory task. The resulting PDM models are comparable in performance to the Volterra models and require significantly less representational and implementational overhead.

This paper presents a novel modeling approach for neural systems with point-process inputs and outputs (binary time-series of 0's and 1's) that utilizes Boolean operators of modulo-2 multiplication and addition, corresponding to the logical AND and OR operations respectively. The form of the employed mathematical model is akin to a "Boolean-Volterra" model that contains the product terms of all relevant input lags in a hierarchical order, where terms of order higher than first represent nonlinear interactions among the various lagged values of each input point-process or among lagged values of various inputs (if multiple inputs exist) as they reflect on the output. The coefficients of this Boolean model are also binary variables that indicate the presence or absence of the respective term in each specific model/system. Simulations are used to explore the properties of such models and the feasibility of accurate estimation of such models from short data-records in the presence of noise (i.e. spurious spikes). The results demonstrate the feasibility of obtaining reliable estimates of such models, even in the presence of considerable noise in the input and/or output, thus making the proposed approach an attractive candidate for modeling neural systems in a practical context.