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Edited by: Shaohui Wang, Louisiana College, United States

Reviewed by: Daya Shankar Gupta, Camden County College, United States; Keke Wang, Embry-Riddle Aeronautical University, Prescott, United States

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) and the copyright owner(s) 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.

High-frequency firing activity can be induced either naturally in a healthy brain as a result of the processing of sensory stimuli or as an uncontrolled synchronous activity characterizing epileptic seizures. As part of this work, we investigate how logic circuits that are engineered in neurons can be used to design spike filters, attenuating high-frequency activity in a neuronal network that can be used to minimize the effects of neurodegenerative disorders such as epilepsy. We propose a reconfigurable filter design built from small neuronal networks that behave as digital logic circuits. We developed a mathematical framework to obtain a transfer function derived from a linearization process of the Hodgkin-Huxley model. Our results suggest that individual gates working as the output of the logic circuits can be used as a reconfigurable filtering technique. Also, as part of the analysis, the analytical model showed similar levels of attenuation in the frequency domain when compared to computational simulations by fine-tuning the synaptic weight. The proposed approach can potentially lead to precise and tunable treatments for neurological conditions that are inspired by communication theory.

Seizure dynamics with either spontaneous and recurrent profiles can occur even in healthy patients during the processing of sensory stimuli or it could manifest itself as an uncontrolled synchronous neural activity in large areas of the brain (Jirsa et al.,

The development of techniques for the treatment of this type of neurodegenerative disorder is challenging not only due to the complexity of the brain function and structure but also as a result of the invasiveness and discomfort caused by today's most common neurostimulation or surgery approaches (Rolston et al.,

Previous studies on the firing response of neurons have investigated the filtering capabilities either due to realistic synaptic dynamics (Brunel et al.,

The manipulation of cellular activity, such as neuronal spiking activity, using molecules complexes to mimic logic gates and transistors has also been proposed in the literature. One example is the work of Vogels and Abbott (

In this work, we propose a mathematical framework aiming at the interpretation of the filtering capabilities in small populations of neurons that are engineered into a logic circuit (

Engineered neuronal digital logic circuit, where each gate is composed of three neurons and each block _{i}(

The remainder of this paper is as follows, section 2.1 briefly describes how neurons differ between each other and how they communicate with one another. In section 2.2, we explain how neurons can function as non-linear electronic circuits based on the seminal work of Hodgkin and Huxley (

To be able to synthetically implement complex functions inside the brain, we must control how the neurons exchange information using the propagation of action potentials inside a network of neurons. The number of excitatory and inhibitory connections between neurons determines the spatio-temporal dynamics of the action potentials propagation (Zhou et al.,

We aim to investigate the neuronal and synaptic properties in constructing logic circuits that perform the filtering of spikes in small populations from the somatosensory cortex. The cortex is responsible for most of the signal processing performed by the brain and comprises a rich variety of morpho-electrical types of neuronal and non-neuronal cells. We will take into account these characteristics in the construction of our mathematical framework that is used to design the circuits.

Neurons are divided into three main parts: dendrites, soma, and axon. Dendrites receive stimuli from other cells and the way these dendritic trees are projected onto neighboring neurons in a network helps to classify neuron morphological types. The axon passes stimuli forward to cells connected down the network through its axon terminals and the soma is the main body of the neuron. Each neuron's response to a stimulus will dictate the electrophysiological neuron type. The soma is where most proteins and genes are produced and where stimuli are generated and fired down the axon.

Besides the way dendrites are projected, the proteins and genes that neurons express and their morphological and electrophysiological characteristics are important for the classification of different types of neurons. One of the most comprehensive works on neuronal modeling, by Markram et al. (

Even though all neurons used in this work can assume different morphological structure, it is exactly by analyzing their axonal and dendritic ramification that we can have a good enough categorization of their respective morphological types. Regardless of their types, neurons in the cortical layer are considered of small sizes (8 - 16μm). Furthermore, inhibitory neurons can be better identified by their axonal features while excitatory neurons can be more easily classified based on their dendritic features (Markram et al.,

The cerebral cortex comprises six distinguished horizontal layers of neurons, with each layer having particular characteristics such as cell density and type, layer size, and thickness. This horizontal configuration is also known as a “laminar” organization, where the layers are identified as (1) Molecular layer, which contains only a few scattered neurons and consists mostly of glial cells and axonal and dendritic connections of neurons from other layers; (2) External granular layer, containing several stellate and small pyramidal neurons; (3) Pyramidal layer, contains non-pyramidal and pyramidal cells of small and medium sizes; (4) Inner granular layer, predominantly populated with stellate and pyramidal cells, this is the target of thalamic inputs; (5) Ganglionic layer, containing large pyramidal cells that establish connections with subcortical structures; and (6) Multiform layer, populated by just a few large pyramidal neurons and a good amount of multiform neurons, which sends information back to the thalamus. All layers may contain inter-neurons bridging two different brain regions.

The neurons are not just stacked one on top of another suggesting a horizontal organization, indeed vertical connections are also found in between the neurons from either the same or different layers. This allows another type of classification known as mini-columns (also called, micro-columns) with a diameter of 30–50 μm and when activated by peripheral stimuli, they are seen as macro-columns, with a diameter of 0.4–0.5 mm (Peters,

The communication between a pair of neurons is done through the diffusion of neurotransmitters in the synaptic cleft; this process is triggered by an electrical impulse reaching the axon terminals of the transmitting cell characterizing an electrochemical signaling process known as the

Schematic of a

In an excitatory synapse, the membrane potential of the post-synaptic cell, which rests at approximately −65 mV, will start depolarizing itself until it reaches a threshold,

The main structures of a neuron, previously mentioned in section 2.1.1, can assume different shapes and spatial structures that play an important role in determining its input and output relationship. By sectioning the neuron into several compartment models, we are able to account for the influence that individual compartments have on the communication process of the neuron. Even though we consider the same value of resting potential for all compartments of the cell, there is some discussion on whether different compartments have different potentials when at rest (Hu and Bean,

We aim to develop a transfer function for the neuron-spike response, or output [

As aforementioned in section 1, neurons can perform spike filtering tasks either by manipulating ionic conductances, such as sodium and potassium conductances, from within the cell (Fortune and Rose,

Hodgkin-Huxley (HH) model:

_{Na} and _{K} and the leak channel by the linear conductance _{l}. The membrane capacitance is proportional to the surface area of the neuron and, along with its resistance, dictates how fast its potential responds to the ionic flow. The ratio between intra- and extra-cellular ions define the reversal potentials _{Na, K, l} establishing a gradient that will drive the flow of ions (Barreto and Cressman,

When an external stimulus, _{ext}, is presented, it triggers either the activation or inactivation of the ionic channels that allow the exchange of ions that result in depolarization (or hyperpolarization when inhibitory) of the membrane of the cell. These dynamics are modeled as

where _{x} are the ionic currents where

where

in which the values of the rate constants α_{i} and β_{i} for the

The membrane capacitance is proportional to the size of the cell, and on the other hand, the bigger the cell diameter, the lower the spontaneous firing rate (Sengupta et al.,

In order to derive a transfer function for the Hodgkin-Huxley model, we must consider each neuron as a system that is linear and time-invariant (LTI). If the system is non-linear, then a linearization process should be done before any frequency analysis is performed. For a more detailed analysis on the procedures for linearization of the Hodgkin-Huxley model, the reader is referred to Koch (

The linearization process requires that we reconsider the electronic components in each neuron compartment to adequately eliminate trivial relationships. Membranes with specific types of voltage- and time-dependent conductances can behave as if they had inductances even though neurobiology does not possess any coil-like elements. This modification will transform the behavior of non-linear components toward linearization, resulting in a proportional relationship between the voltage and current changes (Koch,

Every linearization process is performed for small variations around a fixed point, hereafter denominated by δ, and in the case of the Hodgkin-Huxley model, this fixed point should be the steady-state (resting state) of the system. Because the sodium activation generates a current component that flows in an opposite direction compared to that of a passive current, the branch concerning the sodium activation should have components with negative values while the branches regarding potassium activation and sodium inactivation should have components with positive values (Sabah and Leibovic, _{n}, _{m}, and _{h} are the conductances of the inductive branches connected in series with their respective inductances _{n}, _{m}, and _{h} derived from the linearization process and _{T} = _{L} + _{K} + _{Na} is the total pure membrane conductance.

Hodgkin-Huxley linear circuit model representation.

Let us consider the membrane potential deviation, δ

where δ_{l,Na,K} are current variations at any given steady-state and can be defined as

where _{K,Na} are pure conductances of potassium and sodium and _{L} the pure leak conductance expressed as

where ḡ_{K,Na} are the maximum attainable conductances, and δ

as a function of the derivative of the rate constants α_{n,m,h} and β_{n,m,h}, and _{∞}, _{∞}, and _{∞} are the steady-state values of

and the conductances, _{n,m,h}, and inductances, _{n,m,h}, of the inductive branches are defined as

Each channel has a probability of being open which represents the fraction of gates in that channel that are in the permissive state (Gerstner et al., _{n,m,h} and their respective inductances _{n,m,h} which are functions of the rate constants representing the transition from permissive to non-permissive state, α(^{−1}, to eventually reach a steady-state value, α_{∞} and β_{∞} (Koslow and Subramaniam,

Borrowing concepts from systems theory such as frequency analysis of LTI systems, as a standard procedure for the analysis of linear differential equations as simpler algebraic expressions (see Nise,

Impedance relationships for capacitors, resistors, and inductors.

Therefore, the relationship between the output and the input of the system in the frequency domain is expressed as

where

Now, denoting γ = _{T}+_{n}+_{m}+_{h} and

For frequency response analysis, we observe the behavior of ^{−1} is the gain,

Given the transfer function for a neural compartment in the previous section, we now progress toward a transfer function for the spike filter. The filter is comprised of neurons that are particularly chosen to have a network that will behave as a digital gate and a small population that will behave as a circuit that implements the filter. Our aim is to capture the relationship between compartments as well as neuron connections so we can build a transfer function for the filter while considering neuron connection variables (synaptic conductance and synaptic weight) that allow easy reconfiguration of the filtering process. The linearization process combined with the analysis of the neuron communications is the driver of the filtering process, which also allows the derivation of a filter transfer function which is detailed below.

Synthetic biology is the technology that allows the control of the neurons' internal process in order to construct non-natural activity and functioning of neurons, e.g., logic gates (Larouche and Aguilar,

Given that several factors such as connection probability, type of cell, and different numbers of compartments (as discussed in section 2.3.2) among different types of neurons may influence its gating capabilities. This variation on the quantity of compartments could also lead to variations on periods for the action potential to reach the post-synaptic terminals and start the synapse process. Furthermore, cells with bigger sizes of soma may take more time and amount of stimuli to reach threshold for action potential initiation (Sengupta et al., _{i}(

Neurons are very complex structures with numerous ramifications and several factors that contribute to their highly non-linear dynamism. Aiming to make the comprehension of such a complex electrical behavior easier, one employs a widely used technique called “compartmental modeling.” Since different neurons have different morphologies, the mechanism of determining the number of compartments will be based on estimating the length of a specific neuronal structure. For instance, a varying length of axon, which will reflect in different quantities of compartment in series, where we will have a fixed size for each segment of the axon representing one compartment. This is a very natural and elegant way to model dynamic systems as multiple interconnected compartments where each compartment is described by its own set of equations, carrying the influence of one compartment to the next reproducing the behavior of the whole neuron.

Observing the neuron as a set of compartments described by transfer functions equivalent to that of (35), the neuronal morphology of a pyramidal cell, as illustrated in

Compartmental neuron representation:

Every single compartment, each represented by one transfer function, is grouped in trees of three cells (

with symbols defined previously, and a new parameter ζ_{i} describing the synaptic weight for the _{i} acts as a tunable gain for the neurons.

Using the parameters from (Mauro et al.,

and its frequency response (Bode plot) for the relevant range of frequencies in our applications (Wilson et al.,

Bode plots:

Let us now observe three cases concerning the choice of ζ_{i} values. In the first case, we keep all of them at unity and consider it our base case for this part of the analysis (and to keep it aligned with the rest of the paper, we call it _{3} and ζ_{6}, which corresponds to the manipulation of the output cell for the two input gates in _{9} and leaving everything else intact. Finally, in the third case, we manipulate the output cell of the last gate by halving its synaptic conductance (_{9B} = 1, ζ_{9A} = 2, and ζ_{9C} = 1/2, respectively. Since the tunable gain ζ_{9} of the gate _{9}, is the tunable gain of the whole system

Linear model filter analysis

Summary of elements described in the proposed model.

Membrane capacitance | |

_{Na},_{K},_{l} |
Sodium, potassium, and leak conductances |

_{Na},_{K},_{l} |
Sodium, potassium, and leak reversal potentials |

_{ext} |
External stimulus |

_{Na},_{K},_{l} |
Ionic current for the sodium, potassium, and leak channels |

Membrane potential | |

Sodium activation and inactivation variables | |

Potassium activation variable | |

α,β | Rate constants for |

δ | Small variation around the steady-state |

_{T} |
Total pure conductance |

_{Na},_{K},_{L} |
Sodium, potassium, and leak pure conductances |

ḡ_{Na},ḡ_{K},ḡ_{l} |
Maximum attainable sodium, potassium and leak conductances |

_{∞},_{∞},_{∞} |
Steady-state values of |

_{m},_{h},_{n} |
Conductances of the inductive branches |

_{m},_{h},_{n} |
Inductances of the ionic paths |

Transfer function of the filter | |

_{0} |
Gain, selectivity, and peak frequency of the filter |

ζ | Synaptic weight |

Alternatively, as we suggested earlier, a single transfer function of a compartment serves as an approximation of the entire system due to the effects of repeated bandpass filtering in _{10}2 ≈ 6 dB in case of halving/doubling the synaptic weight. Since the filter is of a band-passing nature, it is only natural that, around the resonant frequency, lower and higher frequency amplitudes should be ideally attenuated toward zero. Thus, it is worth mentioning that in both cases depicted here, the part of the frequency response with the cusp is at very low frequencies, so it is not visible in the relevant part of the spectrum. As such, the filter behaves as a low pass filter for all practical considerations.

In this section, we discuss the simulation results concerning the reconfigurable logic gates as well as the circuits. For all simulations, intrinsic parameters of the cell were kept at their default values (such as the length and diameter of each of their compartments) meaning that nothing concerning their morphological properties was changed, the spike trains fed to the input of the circuits followed a ^{1}

In this work, we call “reconfigurable” logic gates, the gates that work by changing the synaptic weight between the connections of both input cells with the output cell in a neuronal logic gate structure. Aiming to measure individual gate accuracy, the spike trains in the inputs were randomly produced but we control their frequency variation, in other words, for each simulation, the frequency at all inputs was the same and any change in the frequency was performed for all inputs of the gates meaning that none of the simulations account for different frequency values between different inputs in a single simulation. The accuracy is a simple but powerful measure for the performance of the gates, with which we intend to analyze the effects of the dynamics of the cell on the output of the circuit when comparing this output with the ideal response of the circuit derived from its truth-table. The accuracy is calculated according to the following equation (Hanisch and Pierobon,

where _{Y,E[Y]} is the probability of _{Y,E[Y]} resembles the conditional probabilities in a binary symmetric channel (BSC). Thus, _{0,0} = 1 − _{1,0}, and _{0,1} = 1 − _{1,1}. It is possible to calculate _{1,1}, for instance, by counting the number of bits there are for each input-output combination. In other words, considering _{i,j} the number of times a bit _{1,1} = _{1,1}/(_{1,1} + _{0,1}).

Given the objective of obtaining a behavior similar to an OR gate, the synaptic weight should be set to 0.06 μS, meaning that the pre-synaptic stimuli will drive a higher influence on the depolarization of the post-synaptic cell. On the other hand, for an AND behavior, the weight is set to 0.03 μS, which reduces the influence of a single spike and look to a response of the post-synaptic neuron only when two spikes arrive very close to each other in terms of time. This is conducted so we have acceptable levels of accuracy when compared to the expected outputs of the gate.

Analysis on reconfigurable logic gates with neurons of types

Once the reconfigurable behavior of the gates is assessed, they are connected to other gates to form a logic circuit. The performance is measured employing a ratio (frequency response), i.e., the number of spikes (bits “1”) in the output divided by the nominal input frequency, in Hertz. This ratio is also known as the magnitude, or gain when evaluating the data in decibels. Following the approach for individual gates, the inputs are random and the frequency is increased uniformly. Since the gates showed similar accuracy when increasing the input frequency, we picked the one analyzed in

Effects of dynamic changes to the synaptic weight in circuits A, B, and C;

In the non-linear case of the system, the filtering is even better than what the linear model would promise, i.e., the suppression of unwanted frequencies is better due to superexponential decay. Let us compare

Now, let us consider

where ν_{c} is the cut-off frequency and ν_{f} is the last evaluated frequency (in this relationship, the lower the value, the more efficient the filter is). Since, in terms of magnitude, a frequency band when cut by an ideal filter should be attenuated toward negative infinity (−∞), we have to pick a limit for the calculation of the area under the curves. In our case, after a visual inspection, the baseline for calculation chosen was −25 dB, because this is the closest integer value to the lowest values of magnitude.

Counter-efficiency of the circuits when compared to ideal filters (the lower the value, the better the filter's performance).

This shift in performance may allow us to control which type of circuit we want to activate inside the brain depending on which activity the subject is performing at the time, e.g., being awake or being asleep. These changes may be induced by the intake of specific drugs that alter synaptic properties in a neuronal connection.

Parallel between Magnitude (dB) and accuracy of circuits

The results suggest that, by increasing the number of gates in cascade, we have to deal with attenuation in the network due to propagation caused by specific characteristics of the cell, such as the connection probability; hence, the more gates in cascade the worse the performance of the circuit. Also, even though the ratio keeps going downwards, at some point, the accuracy will start to shoot up. With careful evaluation, the dip in the accuracy along mid-range frequencies is very low in terms of scale, showing a difference of only around 0.03 on the values of accuracy.

Synaptic weight plays a role in the influence of the pre-synaptic stimuli and its impact on the post-synaptic neuron and has a value proportional to the synaptic conductance (Gardner,

With our model, we have mainly investigated the attenuation on the spiking frequency for three different types of circuits in which we decrease the number of OR gates by replacing them with AND gates. We were also able to have the fine-tuning synaptic properties showing a difference of around 5 dB in performance between the curves in

Our results, therefore, suggest that neuronal logic circuits can be used to construct also digital filters, filtering abnormal high-frequency activity which can have many sources including neurodegenerative diseases. A metric of counter-efficiency was also proposed, which should show how far apart the real results are from the ideal cases. We found that frequency bands were found to be of optimal value for different types of circuits such as 60 Hz for circuit C, 100 Hz for circuit B, and 120 Hz for circuit A, as shown in

The envisioned application of the proposed mathematical framework is for

In this work, we proposed a reconfigurable spike filtering design using neuronal networks that behave as a digital logic circuit. This approach requires the cells to be sensitive to modifications through chemicals delivered through several proposed methods available in the literature. From the Hodgkin-Huxley action potential model we developed a mathematical framework to obtain the transfer function of the filter. This required a linearization of the Hodgkin-Huxley model that changes the cable theory simplification for each cell compartment. To evaluate the system, we have used our transfer function as well as the NEURON simulator to show how the frequency of operation, logic circuit configuration as well as logic circuit size can affect the accuracy and efficiency of the signal propagation. We observed that all-ANDs circuit produces more accurate results concerning their truth-table when compared to all-ORs. In addition, the results show that each digital logic circuit is also reconfigurable in terms of cut-off frequency of the filter, by manipulating the types of gates in the last layer of the circuit.

We believe the proposed filter design and its mathematical framework will contribute to synthetic biology approaches for neurodegenerative disorders such as epilepsy, by showing how the control of cellular communication inside a small population can affect the propagation of signals. For future work, we plan the use of non-neuronal cells, e.g. astrocytes, for the control of gating operations and the assessment of neuronal filtering capabilities at a network level. Treatment techniques based on this method can be a radical new approach to reaching precision and adaptable outcomes, inspired from electronic engineering as well as communication engineering. Such techniques could tackle at a single-cell level, neurons affected by seizure-induced high-frequency firing or bypass neurons that have been affected by a disease-induced neuronal death and degeneration, thus keeping the neuronal pathway working at a performance as optimal as possible.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

GA performed the simulations and wrote the manuscript. HS performed the control-theoretic analysis. GA, HS, and MB performed the data analysis. SB, NM, MB, and MW led the work development. All authors contributed to manuscript writing and revision. All authors also have read and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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