A nonlinear autoregressive Volterra model of the Hodgkin-Huxley equations

Deep evolutionary fusion neural network: a new prediction standard for infectious disease incidence rates

BACKGROUND

Previously, many methods have been used to predict the incidence trends of infectious diseases. There are numerous methods for predicting the incidence trends of infectious diseases, and they have exhibited varying degrees of success. However, there are a lack of prediction benchmarks that integrate linear and nonlinear methods and effectively use internet data. The aim of this paper is to develop a prediction model of the incidence rate of infectious diseases that integrates multiple methods and multisource data, realizing ground-breaking research.

RESULTS

The infectious disease dataset is from an official release and includes four national and three regional datasets. The Baidu index platform provides internet data. We choose a single model (seasonal autoregressive integrated moving average (SARIMA), nonlinear autoregressive neural network (NAR), and long short-term memory (LSTM)) and a deep evolutionary fusion neural network (DEFNN). The DEFNN is built using the idea of neural evolution and fusion, and the DEFNN + is built using multisource data. We compare the model accuracy on reference group data and validate the model generalizability on external data. (1) The loss of SA-LSTM in the reference group dataset is 0.4919, which is significantly better than that of other single models. (2) The loss values of SA-LSTM on the national and regional external datasets are 0.9666, 1.2437, 0.2472, 0.7239, 1.4026, and 0.6868. (3) When multisource indices are added to the national dataset, the loss of the DEFNN + increases to 0.4212, 0.8218, 1.0331, and 0.8575.

CONCLUSIONS

We propose an SA-LSTM optimization model with good accuracy and generalizability based on the concept of multiple methods and multiple data fusion. DEFNN enriches and supplements infectious disease prediction methodologies, can serve as a new benchmark for future infectious disease predictions and provides a reference for the prediction of the incidence rates of various infectious diseases.

Nonlinear dynamical modeling of neural activity using volterra series with GA-enhanced particle swarm optimization algorithm

In order to improve the modeling performance of Volterra sequence for nonlinear neural activity, in this paper, a new optimization algorithm is proposed to identify Volterra sequence parameters. Algorithm combines the advantages of particle swarm optimization (PSO) and genetic algorithm (GA) improve the performance of the identification of nonlinear model parameters from rapidity and accuracy. In the modeling experiments of neural signal data generated by the neural computing model and clinical neural data set in this paper, the proposed algorithm shows its excellent potential in nonlinear neural activity modeling. Compared with PSO and GA, the algorithm can achieve less identification error, and better balance the convergence speed and identification error. Further, we explore the influence of algorithm parameters on identification efficiency, which provides possible guiding significance for parameter setting in practical application of the algorithm.

Model Predictive Control for Seizure Suppression Based on Nonlinear Auto-Regressive Moving-Average Volterra Model

This article investigates a closed-loop brain stimulation method based on model predictive control strategy to suppress epileptic seizures. A neural mass model (NMM), exhibiting the normal and various epileptic seizures by changing physiologically meaningful parameters, is used as a black-box model of the brain. Based on system identification, an auto-regressive moving-average Volterra model is established to reveal the relationship between stimulation and neuronal responses. Then, the model predictive control strategy is implemented based the Volterra model, which can generate an optimal stimulation waveform to eliminate epileptiform waves. The computational simulation results indicate the proposed closed-loop control strategy can optimize the stimulation waveform without particular knowledge of the physiological properties of the system. The robustness of the proposed control strategy to system disturbances makes it more appropriate for future clinical application.

Quantifying the Nonlinear Interaction in the Nervous System Based on Phase-Locked Amplitude Relationship

OBJECTIVE

This paper introduces the Cross-frequency Amplitude Transfer Function (CATF), a model-free method for quantifying nonlinear stimulus-response interaction based on phase-locked amplitude relationship.

METHOD

The CATF estimates the amplitude transfer from input frequencies at stimulation signal to their harmonics/intermodulation at the response signal. We first verified the performance of CATF in simulation tests with systems containing a static nonlinear function and a linear dynamic, i.e., Hammerstein and Wiener systems. We then applied the CATF to investigate the second-order nonlinear amplitude transfer in the human proprioceptive system from the periphery to the cortex.

RESULT

The simulation demonstrated that the CATF is a general method, which can well quantify nonlinear stimulus-response amplitude transfer for different orders of nonlinearity in Wiener or Hammerstein system configurations. Applied to the human proprioceptive system, we found a complicated nonlinear system behavior with substantial amplitude transfer from the periphery stimulation to cortical response signals in the alpha band. This complicated system behavior may be associated with the nonlinear behavior of the muscle spindle and the dynamic interaction in the thalamocortical radiation.

CONCLUSION

This paper provides a new tool to identify nonlinear interaction in the nervous system.

SIGNIFICANCE

The results provide novel insight of nonlinear dynamics in the human proprioceptive system.

Nonlinear predictive control for adaptive adjustments of deep brain stimulation parameters in basal ganglia-thalamic network

The efficacy of deep brain stimulation (DBS) for Parkinson's disease (PD) depends in part on the post-operative programming of stimulation parameters. Closed-loop stimulation is one method to realize the frequent adjustment of stimulation parameters. This paper introduced the nonlinear predictive control method into the online adjustment of DBS amplitude and frequency. This approach was tested in a computational model of basal ganglia-thalamic network. The autoregressive Volterra model was used to identify the process model based on physiological data. Simulation results illustrated the efficiency of closed-loop stimulation methods (amplitude adjustment and frequency adjustment) in improving the relay reliability of thalamic neurons compared with the PD state. Besides, compared with the 130Hz constant DBS the closed-loop stimulation methods can significantly reduce the energy consumption. Through the analysis of inter-spike-intervals (ISIs) distribution of basal ganglia neurons, the evoked network activity by the closed-loop frequency adjustment stimulation was closer to the normal state.

Methodology of Recurrent Laguerre-Volterra Network for Modeling Nonlinear Dynamic Systems

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 l₁ 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.

Principal dynamic mode analysis of neural mass model for the identification of epileptic states

The detection of epileptic seizures in Electroencephalography (EEG) signals is significant for the diagnosis and treatment of epilepsy. In this paper, in order to obtain characteristics of various epileptiform EEGs that may differentiate different states of epilepsy, the concept of Principal Dynamic Modes (PDMs) was incorporated to an autoregressive model framework. First, the neural mass model was used to simulate the required intracerebral EEG signals of various epileptiform activities. Then, the PDMs estimated from the nonlinear autoregressive Volterra models, as well as the corresponding Associated Nonlinear Functions (ANFs), were used for the modeling of epileptic EEGs. The efficient PDM modeling approach provided physiological interpretation of the system. Results revealed that the ANFs of the 1st and 2nd PDMs for the auto-regressive input exhibited evident differences among different states of epilepsy, where the ANFs of the sustained spikes' activity encountered at seizure onset or during a seizure were the most differentiable from that of the normal state. Therefore, the ANFs may be characteristics for the classification of normal and seizure states in the clinical detection of seizures and thus provide assistance for the diagnosis of epilepsy.

Understanding spike-triggered covariance using Wiener theory for receptive field identification

Receptive field identification is a vital problem in sensory neurophysiology and vision. Much research has been done in identifying the receptive fields of nonlinear neurons whose firing rate is determined by the nonlinear interactions of a small number of linear filters. Despite more advanced methods that have been proposed, spike-triggered covariance (STC) continues to be the most widely used method in such situations due to its simplicity and intuitiveness. Although the connection between STC and Wiener/Volterra kernels has often been mentioned in the literature, this relationship has never been explicitly derived. Here we derive this relationship and show that the STC matrix is actually a modified version of the second-order Wiener kernel, which incorporates the input autocorrelation and mixes first- and second-order dynamics. It is then shown how, with little modification of the STC method, the Wiener kernels may be obtained and, from them, the principal dynamic modes, a set of compact and efficient linear filters that essentially combine the spike-triggered average and STC matrix and generalize to systems with both continuous and point-process outputs. Finally, using Wiener theory, we show how these obtained filters may be corrected when they were estimated using correlated inputs. Our correction technique is shown to be superior to those commonly used in the literature for both correlated Gaussian images and natural images.

Hippocampal closed-loop modeling and implications for seizure stimulation design

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.

Principal dynamic mode analysis of the Hodgkin-Huxley equations

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.

Volterra dendritic stimulus processors and biophysical spike generators with intrinsic noise sources

We consider a class of neural circuit models with internal noise sources arising in sensory systems. The basic neuron model in these circuits consists of a dendritic stimulus processor (DSP) cascaded with a biophysical spike generator (BSG). The dendritic stimulus processor is modeled as a set of nonlinear operators that are assumed to have a Volterra series representation. Biophysical point neuron models, such as the Hodgkin-Huxley neuron, are used to model the spike generator. We address the question of how intrinsic noise sources affect the precision in encoding and decoding of sensory stimuli and the functional identification of its sensory circuits. We investigate two intrinsic noise sources arising (i) in the active dendritic trees underlying the DSPs, and (ii) in the ion channels of the BSGs. Noise in dendritic stimulus processing arises from a combined effect of variability in synaptic transmission and dendritic interactions. Channel noise arises in the BSGs due to the fluctuation of the number of the active ion channels. Using a stochastic differential equations formalism we show that encoding with a neuron model consisting of a nonlinear DSP cascaded with a BSG with intrinsic noise sources can be treated as generalized sampling with noisy measurements. For single-input multi-output neural circuit models with feedforward, feedback and cross-feedback DSPs cascaded with BSGs we theoretically analyze the effect of noise sources on stimulus decoding. Building on a key duality property, the effect of noise parameters on the precision of the functional identification of the complete neural circuit with DSP/BSG neuron models is given. We demonstrate through extensive simulations the effects of noise on encoding stimuli with circuits that include neuron models that are akin to those commonly seen in sensory systems, e.g., complex cells in V1.

On parsing the neural code in the prefrontal cortex of primates using principal dynamic modes

Nonlinear modeling of multi-input multi-output (MIMO) neuronal systems using Principal Dynamic Modes (PDMs) provides a novel method for analyzing the functional connectivity between neuronal groups. This paper presents the PDM-based modeling methodology and initial results from actual multi-unit recordings in the prefrontal cortex of non-human primates. We used the PDMs to analyze the dynamic transformations of spike train activity from Layer 2 (input) to Layer 5 (output) of the prefrontal cortex in primates performing a Delayed-Match-to-Sample task. The PDM-based models reduce the complexity of representing large-scale neural MIMO systems that involve large numbers of neurons, and also offer the prospect of improved biological/physiological interpretation of the obtained models. PDM analysis of neuronal connectivity in this system revealed "input-output channels of communication" corresponding to specific bands of neural rhythms that quantify the relative importance of these frequency-specific PDMs across a variety of different tasks. We found that behavioral performance during the Delayed-Match-to-Sample task (correct vs. incorrect outcome) was associated with differential activation of frequency-specific PDMs in the prefrontal cortex.