Volterra representation enables modeling of complex synaptic nonlinear dynamics in large-scale simulations

Bridging Hierarchies in Multi-Scale Models of Neural Systems: Look-Up Tables Enable Computationally Efficient Simulations of Non-linear Synaptic Dynamics

Synapses are critical actors of neuronal transmission as they form the basis of chemical communication between neurons. Accurate computational models of synaptic dynamics may prove important in elucidating emergent properties across hierarchical scales. Yet, in large-scale neuronal network simulations, synapses are often modeled as highly simplified linear exponential functions due to their small computational footprint. However, these models cannot capture the complex non-linear dynamics that biological synapses exhibit and thus, are insufficient in representing synaptic behavior accurately. Existing detailed mechanistic synapse models can replicate these non-linear dynamics by modeling the underlying kinetics of biological synapses, but their high complexity prevents them from being a suitable option in large-scale models due to long simulation times. This motivates the development of more parsimonious models that can capture the complex non-linear dynamics of synapses accurately while maintaining a minimal computational cost. We propose a look-up table approach that stores precomputed values thereby circumventing most computations at runtime and enabling extremely fast simulations for glutamatergic receptors AMPAr and NMDAr. Our results demonstrate that this methodology is capable of replicating the dynamics of biological synapses as accurately as the mechanistic synapse models while offering up to a 56-fold increase in speed. This powerful approach allows for multi-scale neuronal networks to be simulated at large scales, enabling the investigation of how low-level synaptic activity may lead to changes in high-level phenomena, such as memory and learning.

Modeling Nonlinear Synaptic Dynamics: A Laguerre-Volterra Network Framework for Improved Computational Efficiency in Large Scale Simulations

Synapses are key components in signal transmission in the brain, often exhibiting complex non-linear dynamics. Yet, they are often crudely modelled as linear exponential equations in large-scale neuron network simulations. Mechanistic models that use detailed channel receptor kinetics more closely replicate the nonlinear dynamics observed at synapses, but use of such models are generally restricted to small scale simulations due to their computational complexity. Previously, we have developed an ``input-output'' (IO) synapse model using the Volterra functional series to estimate nonlinear synaptic dynamics. Here, we present an improvement on the IO synapse model using the extbf{Laguerre-Volterra network (LVN) framework. We demonstrate that utilization of the LVN framework helps reduce memory requirements and improves the simulation speed in comparison to the previous iteration of the IO synapse model. We present results that demonstrate the accuracy, memory efficiency, and speed of the LVN model that can be extended to simulations with large numbers of synapses. Our efforts enable complex nonlinear synaptic dynamics to be modelled in large-scale network models, allowing us to explore how synaptic activity may influence network behavior and affects memory, learning, and neurodegenerative diseases.

A Glutamatergic Spine Model to Enable Multi-Scale Modeling of Nonlinear Calcium Dynamics

In synapses, calcium is required for modulating synaptic transmission, plasticity, synaptogenesis, and synaptic pruning. The regulation of calcium dynamics within neurons involves cellular mechanisms such as synaptically activated channels and pumps, calcium buffers, and calcium sequestrating organelles. Many experimental studies tend to focus on only one or a small number of these mechanisms, as technical limitations make it difficult to observe all features at once. Computational modeling enables incorporation of many of these properties together, allowing for more complete and integrated studies. However, the scale of existing detailed models is often limited to synaptic and dendritic compartments as the computational burden rapidly increases when these models are integrated in cellular or network level simulations. In this article we present a computational model of calcium dynamics at the postsynaptic spine of a CA1 pyramidal neuron, as well as a methodology that enables its implementation in multi-scale, large-scale simulations. We first present a mechanistic model that includes individually validated models of various components involved in the regulation of calcium at the spine. We validated our mechanistic model by comparing simulated calcium levels to experimental data found in the literature. We performed additional simulations with the mechanistic model to determine how the simulated calcium activity varies with respect to presynaptic-postsynaptic stimulation intervals and spine distance from the soma. We then developed an input-output (IO) model that complements the mechanistic calcium model and provide a computationally efficient representation for use in larger scale modeling studies; we show the performance of the IO model compared to the mechanistic model in terms of accuracy and speed. The models presented here help achieve two objectives. First, the mechanistic model provides a comprehensive platform to describe spine calcium dynamics based on individual contributing factors. Second, the IO model is trained on the main dynamical features of the mechanistic model and enables nonlinear spine calcium modeling on the cell and network level simulation scales. Utilizing both model representations provide a multi-level perspective on calcium dynamics, originating from the molecular interactions at spines and propagating the effects to higher levels of activity involved in network behavior.

Model-Based Analysis of Electrode Placement and Pulse Amplitude for Hippocampal Stimulation

Objective:

The ideal form of a neural-interfacing device is highly dependent upon the anatomy of the region with which it is meant to interface. Multiple-electrode arrays provide a system which can be adapted to various neural geometries. Computational models of stimulating systems have proven useful for evaluating electrode placement and stimulation protocols, but have yet to be adequately adapted to the unique features of the hippocampus.

Methods:

As an approach to understanding potential memory restorative devices, an Admittance Method-NEURON model was constructed to predict the direct and synaptic response of a region of the rat dentate gyrus to electrical stimulation of the perforant path.

Results:

A validation of estimated local field potentials against experimental recordings is performed and results of a bi-linear electrode placement and stimulation amplitude parameter search are presented.

Conclusion:

The parametric analysis presented herein suggests that stimulating electrodes placed between the lateral and medial perforant path, near the crest of the dentate gyrus, yield a larger relative population response to given stimuli.

Significance:

Beyond deepening understanding of the hippocampal tissue system, establishment of this model provides a method to evaluate candidate stimulating devices and protocols.

Mechanism-Based and Input-Output Modeling of the Key Neuronal Connections and Signal Transformations in the CA3-CA1 Regions of the Hippocampus

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.

Development of a Computational Approach/Model to Explore NMDA Receptors Functions

Modern laboratory techniques allow studying NMDA receptors (NMDAR) either anatomically with specific antibodies coupled to sophisticated confocal microscopy, or physiologically by live imaging or electrophysiological techniques. However, NMDARs are not fixed in time and space and changes in their composition and/or distribution on the post-synaptic membrane may significantly impact the synaptic strength and overall function. The computational modeling approach therefore constitutes a complementary tool for investigating the properties of biological systems based on the knowledge provided by the lab experiments.Here, we describe a general computational method aiming at developing kinetic Markov-chain based models of NMDARs subtypes capable of reproducing various experimental results. These models are then used to make predictions on additional (non-obvious) properties and on their role in synaptic function under various physiological and pharmacological conditions. For the purpose of this book chapter, we will focus on the method used to develop a NMDAR model that includes pharmacological site of action of different compounds. Notably, this elementary model can subsequently be included in a neuron model (not described in detail here) to explore the impact of their differential distribution on synaptic functions.

Development of a detailed model of calcium dynamics at the postsynaptic spine of an excitatory synapse

Postsynaptic calcium dynamics play a critical role in synaptic plasticity, but are often difficult to measure in experimental protocols due to their relatively fast rise and decay times, and the small spine dimensions. To circumvent these limitations, we propose to develop a computational model of calcium dynamics in the postsynaptic spine. This model integrates the main elements that participate in calcium concentration influx, efflux, diffusion and buffering. These consist of (i) spine geometry; (ii) calcium influx through NMDA receptors and voltage-dependent calcium channels (VDCC); (iii) calcium efflux with plasma membrane calcium pumps (PMCA) and sodium-calcium exchangers (NCX); (iv) intracellular calcium stores; and (v) calcium buffers. We herein present computational results we obtained and compare them with experimentally measured data, thereby validating the proposed model. Overall the development of such postsynaptic calcium model may help us better understand the intricacies of interplay between the different elements that shape calcium dynamics and impact synaptic plasticity in normal functions and pathologies. This model also constitutes a first step in the development of a nonlinear input-output calcium dynamics model for multi-scale, large scale neuronal simulations.