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arXiv:1906.07570 · Quantum

Quantized Three-Ion-Channel Neuron Model for Neural Action Potentials

Tasio Gonzalez-Raya, E. Solano, Mikel Sanz

Published Jan 20, 2020• 13 citations
arXivHTMLPDF

Abstract

The Hodgkin-Huxley model describes the conduction of the nervous impulse through the axon, whose membrane's electric response can be described employing multiple connected electric circuits containing capacitors, voltage sources, and conductances. These conductances depend on previous depolarizing membrane voltages, which can be identified with a memory resistive element called memristor. Inspired by the recent quantization of the memristor, a simplified Hodgkin-Huxley model including a single ion channel has been studied in the quantum regime. Here, we study the quantization of the complete Hodgkin-Huxley model, accounting for all three ion channels, and introduce a quantum source, together with an output waveguide as the connection to a subsequent neuron. Our system consists of two memristors and one resistor, describing potassium, sodium, and chloride ion channel conductances, respectively, and a capacitor to account for the axon's membrane capacitance. We study the behavior of both ion channel conductivities and the circuit voltage, and we compare the results with those of the single channel, for a given quantum state of the source. It is remarkable that, in opposition to the single-channel model, we are able to reproduce the voltage spike in an adiabatic regime. Arguing that the circuit voltage is a quantum variable, we find a purely quantum-mechanical contribution in the system voltage's second moment. This work represents a complete study of the Hodgkin-Huxley model in the quantum regime, establishing a recipe for constructing quantum neuron networks with quantum state inputs. This paves the way for advances in hardware-based neuromorphic quantum computing, as well as quantum machine learning, which might be more efficient resource-wise.

DOI
10.22331/q-2020-01-20-224

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The Hodgkin-Huxley model describes the conduction of the nervous impulse through the axon, whose membrane's electric response can be described employing multiple connected electric circuits containing capacitors, voltage sources, and conductances. These conductances depend on previous depolarizing membrane voltages, which can be identified with a memory resistive element called memristor. Inspired by the recent quantization of the memristor, a simplified Hodgkin-Huxley model including a single ion channel has been studied in the quantum regime. Here, we study the quantization of the complete Hodgkin-Huxley model, accounting for all three ion channels, and introduce a quantum source, together with an output waveguide as the connection to a subsequent neuron. Our system consists of two memristors and one resistor, describing potassium, sodium, and chloride ion channel conductances, respectively, and a capacitor to account for the axon's membrane capacitance. We study the behavior of both ion channel conductivities and the circuit voltage, and we compare the results with those of the single channel, for a given quantum state of the source. It is remarkable that, in opposition to the single-channel model, we are able to reproduce the voltage spike in an adiabatic regime. Arguing that the circuit voltage is a quantum variable, we find a purely quantum-mechanical contribution in the system voltage's second moment. This work represents a complete study of the Hodgkin-Huxley model in the quantum regime, establishing a recipe for constructing quantum neuron networks with quantum state inputs. This paves the way for advances in hardware-based neuromorphic quantum computing, as well as quantum machine learning, which might be more efficient resource-wise.
The familiar Hodgkin-Huxley (HH) model describes transmission of a nerve impulse through an axon's membrane in terms of the dynamics of connected electrical circuits, featuring capacitors, voltage sources, and memristors. The recent quantization of the memristor's dynamics now allows study of a $q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}m$ version of the HH circuit. The authors reproduce the behavior of the single-ion-channel HH circuit in the quantum regime, and exploring the possibility of this setup to be the building block for bioinspired quantum neural networks. Their results will impact the fields of superconducting devices, neuromorphic quantum computing, and hardware-based solutions for quantum machine learning.
We investigate the effects of a quantum entanglement regime within an ion conducting molecule (ion channel) of the neuronal plasma membrane on the onset dynamics of propagating nerve pulses (action potentials). In particular, we model the onset parameters of the sodium current in the Hodgkin Huxley equation as three similar but independent probabilistic mechanisms which become quantum entangled. The underlying physics is general and can involve entanglement between various degrees of freedom underlaying ion transition states or 'gating states' during conduction, e.g. Na+^++ ions in different channel locations, or different 'affinity' states of ions with atoms lining the sub-regions of the channel protein ('filter-states'). We find that the 'quantum corrected' Hodgkin Huxley equation incorporating entangled systems states can reproduce action potential pulses with the critical onset dynamics observed recently in neocortical neurons in vivo by Naundorf et al. [Nature {\bf 440}, 1060 (20 April 2006)]. Interestingly, the suggested entanglement term can also slow down action potential initiation.
Understanding neural excitability and action potential generation is a cornerstone of neuroscience, relying on the complex interplay of biophysical mechanisms governed by ion channels in neuronal membranes. Traditional approaches, such as the Hodgkin-Huxley model, have provided a robust framework for simulating these processes using classical computational techniques. However, the advent of quantum computing opens new avenues for exploring neuronal dynamics, particularly due to ion movement's inherently quantum mechanical properties at atomic scales. In this study, I investigated the intersection of quantum computing and membrane biophysics, evaluating how quantum algorithms could revolutionize simulations of ion channel behavior and neuronal signaling. I began by analyzing the limitations of classical methods in handling high-dimensional, nonlinear systems. Next, I introduced key quantum computing principles, including qubits, superposition, and entanglement, and their potential applications in solving molecular dynamics and differential equations relevant to neuronal activity. Despite promising theoretical advantages, practical challenges such as qubit coherence, error rates, and hardware scalability must be addressed before quantum models can outperform classical simulations. I also discussed hybrid classical-quantum approaches as a transitional strategy, leveraging the strengths of both paradigms. By bridging neuroscience, biophysics, and quantum information science, I aimed to stimulate interdisciplinary research toward more efficient and accurate models of neural excitability.
The dynamical system corresponding to the Hodgkin-Huxley (HH) neuron contains the control parameter, for instance, the electrical current or another external signal, stimulating the action potential (outcome) in the axon. Choosing the appropriate shape of the control via speed gradient or alternative algorithms one can keep the system imitating a quantum behaviour. The controlled four-dimensional HH system, in this case, involves effects similar to the quantum phase contributions to the computational process. We provide a simple example of the HH-based computational algorithm following the quantum paradigm. The linear chain of two HH neurons emulates the results of the Deutsch – Jozsa algorithm for the searching problem. To reproduce the output effect similar to the contribution of quantum phases the neurons are controlled by one of two alternative feedback versions: target attractor or speed gradient. We invent the successful classical emulation of the Deutch-type quantum algorithm and discuss the pros and cons of both alternative feedback methods. Our approach can open a novel method for the practical realization of quantum algorithms and develop new perspectives for the computational properties of artificial neural networks (ANNs). The possible applications of the proposed algorithms are the modelling of epilepsy in the ANNs and the big data analysis.
We present a theoretical framework for a quantized memristive Leaky Integrate-and-Fire (LIF) neuron, uniting principles from neuromorphic engineering and open quantum systems. Starting from a classical memristive LIF circuit, we apply canonical quantization techniques to derive a quantum model grounded in circuit quantum electrodynamics. Numerical simulations demonstrate key dynamical features of the quantized memristor and LIF neuron in the weak-coupling and adiabatic regime, including memory effects and spiking behavior. Applications of this model to a sound localization benchmark show that it outperforms a phenomenological quantum LIF model as well as a classical LIF. This work establishes a foundational model for quantum neuromorphic computing, offering a pathway toward biologically inspired quantum spiking neural networks and paradigms in quantum machine learning.
It is known that the opening or closing mechanism of a voltage-gated ion channel is triggered by the potential difference across the cell membrane in the nervous system. Based on this picture, we model the ion channel as a nanoscale two-terminal ionic tunneling junction. We apply an external time-varying voltage on the junction to mimic the membrane potential difference in the stimulation of neurons. We derive the non-Markovian quantum Langevin equation from the first principle of quantum mechanics for the ion transport in ion channels, and obtain the ion transport current in terms of quantum tunnelings of ions controlled by the time-varying voltage. We find that the time-varying voltage induces an effective magnetic flux, which causes quantum coherence in ion tunnelings. This effective magnetic flux induces further an oscillatory component to the spectral structure of the ion system, forming a time-dependent quantum memory. Such voltage-induced memory is defined as the active quantum memory, which manifests in the system with a regular oscillatory sideband structure in the ion transport current. The sideband structure demonstrates a multicrossing hysteresis in the current-voltage ( I − V ) curve, responding to the variation of the time-varying voltage (membrane potential difference). We also find that the strength of active quantum memory can be described by the ratio of amplitude and frequency of the time-varying voltage, and can be quantitatively measured through the number of nonzero cross points in the I − V hysteresis and conductance-voltage diagram. Additionally, we explore the temperature dependence of the active quantum memory in such a system. Further, we apply this model to the ion channel system on the biological energy scale. The description of these active quantum memory characteristics provides a quantum mechanical understanding to the underlying mechanism of ion channel dynamics.