Hodgkin-huxley Neuron Model Pdf

[Leonides Suttle]

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The generation of spikes by neurons is energetically a costly process and the evaluation of the metabolic energy required to maintain the signaling activity of neurons a challenge of practical interest. Neuron models are frequently used to represent the dynamics of real neurons but hardly ever to evaluate the electrochemical energy required to maintain that dynamics. This paper discusses the interpretation of a Hodgkin-Huxley circuit as an energy model for real biological neurons and uses it to evaluate the consumption of metabolic energy in the transmission of information between neurons coupled by electrical synapses, i.e., gap junctions. We show that for a single postsynaptic neuron maximum energy efficiency, measured in bits of mutual information per molecule of adenosine triphosphate (ATP) consumed, requires maximum energy consumption. For groups of parallel postsynaptic neurons we determine values of the synaptic conductance at which the energy efficiency of the transmission presents clear maxima at relatively very low values of metabolic energy consumption. Contrary to what could be expected, the best performance occurs at a low energy cost.

MTJ implementation of biorealistic firing behavior. (a) Left, a sketch of the device concept of the proposed MTJ spiking device; right, a sketch of a biological neuron. Both receive the input in terms of currents and produce an output in terms of a voltage variation. In the spintronic device, the potential variation is due to changes in resistance produced by the input current and thermal effects. In the neuron, the potential variation is due to sodium (Na+), potassium (K+), and leakage currents. (b),(c) Comparison of resistance variation of the proposed MTJ device with the potential spiking in a biological neuron, according to the Hodgkin-Huxley model. In (b), from top to bottom, spikes in the resistance through the device, dynamics of the magnetization components, and temperature variation of the device. (c) Example of a numerical simulation of the dynamics of a H-H neuron. It can be observed that the behavior of a sharp firing signal followed by a refractory period is a common feature in both models. For a comparison between the two neurons, see Table 1.

Dependence of (a) the average normalized resistance and (b) temperature as a function of the applied current density. In (a), we identify three regions where (i) the magnetization is strongly biased towards the easy axis in the free layer (nPMA), (ii) there is an auto-oscillation around the easy axis of the free layer (nPMA), and (iii) the magnetization is strongly biased towards the direction of magnetization in the polarizer (p). Firing behavior corresponds to a thermally induced alternation between auto-oscillation and the strongly biased state, as shown in the inset corresponding to the magnetization dynamics of the free layer, as shown in the right panel of (a). Notably, the resistance in this case is given by the mx component, which is represented in blue.

(a) Difference between the H-H model and the LIF model. In the LIF model, we consider pulses of amplitude Δι and period Δτ. (b) Evolution of the device properties in the presence of current pulses. From top to bottom panels, we show the normalized resistance, the evolution of the magnetization components, the profile of the applied current, and the temperature of the device. (c) Time between the first pulse and the first synapse. Different colors represent different pulse sizes, while the x axis represents the lowest-current value in the pulse. To include thermal-induced stochasticity, we simulate each pulse amplitude and size 5 times. (d) Time of the first synapse for a constant current.

Example of a SNN built with the proposed MTJ device. (a) Behavior of a SNN with four neurons. Currents of the two input neurons are set as constants. For each MTJ, we show the behavior of the resistance (upper panel) and input current (lower panel). Input currents of the output neurons are generated according to Eq. (11), with the respective weights, Wmn, shown. (b) Two sets considered for verifying the learning process. (c) Sketch of the feed-forward all-connected SNN. Each pixel is associated with a single input neuron. (d) Evolution of the weight matrix. Starting from a random distribution of weights, after the training process, each figure can be represented by a single output neuron, which has the highest frequency.

A set of techniques for efficient implementation of Hodgkin-Huxley-based (H-H) model of a neural network on FPGA (Field Programmable Gate Array) is presented. The central implementation challenge is H-H model complexity that puts limits on the network size and on the execution speed. However, basics of the original model cannot be compromised when effect of synaptic specifications on the network behavior is the subject of study. To solve the problem, we used computational techniques such as CORDIC (Coordinate Rotation Digital Computer) algorithm and step-by-step integration in the implementation of arithmetic circuits. In addition, we employed different techniques such as sharing resources to preserve the details of model as well as increasing the network size in addition to keeping the network execution speed close to real time while having high precision. Implementation of a two mini-columns network with 120/30 excitatory/inhibitory neurons is provided to investigate the characteristic of our method in practice. The implementation techniques provide an opportunity to construct large FPGA-based network models to investigate the effect of different neurophysiological mechanisms, like voltage-gated channels and synaptic activities, on the behavior of a neural network in an appropriate execution time. Additional to inherent properties of FPGA, like parallelism and re-configurability, our approach makes the FPGA-based system a proper candidate for study on neural control of cognitive robots and systems as well.

Copyright 2014 Yaghini Bonabi, Asgharian, Safari and Nili Ahmadabadi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Conductance-based models are the simplest possible biophysical representation of an excitable cell, such as a neuron, in which its protein molecule ion channels are represented by conductances and its lipid bilayer by a capacitor.

Conductance-based models are based on an equivalent circuit representation of a cell membrane as first put forth by Hodgkin and Huxley (1952). These models represent a minimal biophysical interpretation for an excitable cell in which current flow across the membrane is due to charging of the membrane capacitance, \( I_C \ ,\) and movement of ions across ion channels. In its simplest version, a conductance-based model represents a neuron by a single isopotential electrical compartment, neglects ion movements between subcellular compartments, and represents only ion movements between the inside and outside of the cell. Ion channels are selective for particular ionic species, such as sodium (\(Na\)) or potassium (\(K\)), giving rise to currents \( I_Na \) or \( I_K \ ,\) respectively. Thus, the total membrane current, \( I_m(t) \ ,\) is the sum of the capacitive current and the ionic current,

The leak current, \( I_L \ ,\) approximates the passive properties of the cell. Each ionic current is associated with a conductance (inverse of resistance) and a driving force (battery) which is due to the different concentrations of ions in the intracellular and extracellular media of the cell. Thus,

The parameters in conductance-based models are determined from empirical fits to voltage-clamp experimental data (e.g., see Willms 2002), assuming that the different currents can be adequately separated using pharmacological manipulations and voltage-clamp protocols. As shown in the model formulation, the activation and inactivation variables can be raised to a non-unity integer power, and this is dictated by empirical fits to the data.

Since (i) it is rarely possible to obtain estimates of all parameter values in a conductance-based mathematical model from experimental data alone, and (ii) the model construct is necessarily a simplification of the biological cell, it is important to consider various optimization techniques to help constrain the problem for which the conductance-based model was developed to address.

Conductance-based models are the most common formulation used in neuronal models and can incorporate as many different ion channel types as are known for the particular cell being modeled. A common extension found in many conductance-based models is the inclusion of an equation for calcium dynamics. Ionic currents can be calcium-dependent in addition to voltage-dependent with calcium concentrations being controlled by calcium currents, pumps and exchangers. For example, see section 6.2 in Dayan and Abbott (2001).

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