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Edited by: Julius Georgiou, University of Cyprus, Cyprus

Reviewed by: Jongkil Park, University of California, San Diego, USA; Stavros G. Stavrinides, University of Thessaly, Greece

*Correspondence: Bernabé Linares-Barranco

This article was submitted to Neuromorphic Engineering, a section of the journal Frontiers in Neuroscience

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.

This study firstly presents (i) a novel general cellular mapping scheme for two dimensional neuromorphic dynamical systems such as bio-inspired neuron models, and (ii) an efficient mixed analog-digital circuit, which can be conveniently implemented on a hybrid memristor-crossbar/CMOS platform, for hardware implementation of the scheme. This approach employs 4^{2} memristors and 2

The human nervous system is an intriguing complex system capable of performing intricate tasks using an enormous number of neurons each connected via synapses to several thousand neighboring neurons. Mathematical modeling of biological neural elements such as neurons, synapses, and glial cells has been a long standing active research area (Hodgkin and Huxley,

Special purpose computing architectures have been developed to simulate neurobiological networks and functions using their specially designed software tools for large scale simulations (Ahmadi and Soleimani,

Analog CMOS platform is considered to be a significant choice for direct implementation of neural dynamic functions (Linares-Barranco et al.,

A digital platform is used to realize bio-inspired neural cells. Most digital approaches (Weinstein et al.,

Full custom analog/digital (mixed mode) implementations that comprise low-power, fast analog circuits and programmable, mismatch immune digital circuits. Generally, for this approach, neural computation is performed in the analog domain while the communication of spikes between nervous cells is carried out in the digital domain (Schemmel et al., ^{2} memristors. So, the area consumption is significantly increased with the number of cells. Besides, the circuit provides just one-hot digital values of the dynamical variables, which constrains the networking schemes, so an additional circuit is needed to provide the analog value of the variables.

Our approach is a cellular-based system that discretizes dynamical variables resulting in a cellular phase plane, stores the equilibrium curves in a memristive crossbar-based analog memory block, evaluates the velocity and direction of the vector field in the cells, and tracks the state point in the space using VCOs and pointer registers. This system is a fully reconfigurable general approach capable of implementing a wide range of two dimensional neuromorphic dynamical systems such as FitzHugh-Nagumo (FHN; Fitzhugh,

One promising technology particularly suited for analog and mixed analog-digital computing is based on hybrid circuits that integrate CMOS and memristor devices (Strukov and Likharev,

The implementation constraint has strongly limited the power of dynamical systems in modeling, and neuroscientists have been unable to develop an accurate model. Moreover, a number of behaviors such as special output signal shapes have no elegant analytical description. Our approach significantly alleviates the limitation of computational effort in dynamical functions.

The rest of the paper is organized as follows: In Section 2, we discuss the proposed general mapping scheme. The memristor-crossbar/CMOS based hardware structure of the proposed platform is introduced in Section 3. Section 4 presents implementation of three widely used neuron models called the FHN neuron model (Fitzhugh,

The first step toward developing a memristor-crossbar/CMOS hardware for implementing a reconfigurable neuromorphic dynamical system is drawing a proper mapping to represent target dynamical systems. This step plays a significant role in the whole design process because the proposed mapping ought to satisfy a number of crucial conditions: It ought to have the capability of (1) transforming the relatively complicated equations of neuromorphic dynamical systems into a number of simplified fully implementable equations, (2) obtaining a maximum accuracy considering the available hardware resources for implementation, (3) being applied to a wide range of neuromorphic dynamical systems, and (4) bringing reconfigurability into the hardware by separating the variable features of the neuromorphic dynamical systems (requisite information in phase plane for reconstructing vector field) from the shared features (computations for reproducing time-domain signals), and implementing the variable features on the reconfigurable memristive part of the circuit.

In this section, we propose a cellular mapping that properly satisfies the aforementioned conditions. In Section 2.1, basic concepts of dynamical systems in neuroscience are reviewed, and a general form for two dimensional neuromorphic dynamical systems and its simplified forms are presented. One of the simplified forms, which covers most of the two dimensional neuron models, is the basis of our mapping in the next subsections. In Section 2.2, a two dimensional cellular space is defined and the mapping equations for transforming the target dynamical system from the continuous space to the cellular space is presented. Then, according to the simplified form, a technique for calculating the motion velocity of the state point in each cardinal direction in the cellular space is proposed. Applying this technique to the simplified general form in the cellular space results in a mapping which satisfies first, third and fourth out of four above-mentioned conditions. But the second condition related to the timing and asynchrony ought to be satisfied in order to maximize the accuracy of tracking state point in the cellular space. In Section 2.3, the concept of timing and asynchrony in our cellular approach is explained, the requisite timing condition is introduced, and the final satisfactory timing equations and cell change policy based on the timing parameters are presented.

In computational modeling of neural cell behaviors and phenomena, neuroscientists aim to point out critical features and factors as the system variables and parameters, and draw out deterministic laws governing the evolution of these variables over time (Gerstner and Kistler,

In neuromorphic modeling, two-dimensional dynamical systems are popular because of their capability of phase portrait representation. The overall qualitative dynamics of a system can be easily investigated through the study of the phase portrait of the system including different types of equilibrium points and curves and consequently local velocity vectors in the phase space, and also a geometric representation of special trajectories determining topological behaviors of trajectories in a neighborhood in the phase space. Hence, several neuromorphic models are presented in the general form described as:

where ^{2} represents the location of state point in the plane, and (ẋ, ẏ) (velocity vector) determines the velocity and direction of the motion. According to Equation (1), the terms

In most neuromorphic dynamical systems with the general form of Equation (1) the equilibrium lines are single-valued functions. This implies that one dynamical variable is a single-valued function of another one. Hence, the simplified general form of neuromorphic dynamical systems, ignoring input parameters

Note that our approach supports all four different above-mentioned forms, but considering this fact that most of the popular neuron models such as the FitzHugh-Nagumo (Fitzhugh,

In the first step, we map the continuous phase plane to a cellular plane. As shown in Figure

where

As mentioned, the first simplified general form represents the most common condition in neuromorphic dynamical systems where _{eqx} = _{eqy} =

where _{eqx} and _{eqy} are two arrays of _{eq} array. It will be substantiated that we can evaluate the velocity vector (Ẋ, Ẏ) for all cells in the phase plane just using the equilibrium arrays. Consider a sample point (

Substituting Equation (7) in Equation (8), the velocity vector equation is given by:

As the above equation shows, the velocity vector depends on the equilibrium arrays and a number of predetermined parameters. In other words, the absolute value of the velocity at every state point is proportional to its absolute vertical distance (distance in

According to the previous subsection, the velocity of motion and the motion direction (forward/backward in each dimension) are variable in the cellular phase plane, and can be evaluated using Equation (9). Note that the evaluated velocities are in units of (1∕s) where

Here, with the aim to simplify the timing analysis, we convert the concept of motion velocity to the concept of motion time. Hence, the cellular motion time (the amount of time required for moving one cell) can be obtained by reversing the cellular motion velocity as:

where NF(·) is a function presenting restrictions imposed by hardware implementation that limits the time delay value in a specific boundary. A simple form for this boundary function can be represented as:

where MIN and MAX are lower and upper boundaries. Clearly, our cellular approach is asynchronous, so the state variables are changed asynchronously. Considering this fact, it is crucial to proportionally handle the cellular motion time in one direction when other state variable is changed and consequently causes a change in motion time in both directions. In other words, time handling of the motion in each dimension is independent, but the motion in one dimension influences the motion time in another dimension by changing the vertical distance of the state point from the corresponding equilibrium curve, and it ought to be considered.

Here, we illustrate the time handling procedure for a special case, and then derive a generalized rule to handle the two dimensional timing. Assume that the initial state (

Regarding above explanations, an overall recursive rule can be derived to handle the dimensional motion times for moving from any given cell

where

Thus, when the shorter remaining motion time for the faster state variable is elapsed, the state variable is changed by one cell, and the remaining motion time for the other state variable is updated according to the motion time vector in the new state and using Equation (13). In the following sections, it is qualitatively and quantitatively shown that the dynamical behaviors of a given two dimensional dynamical system described generally by Equation (2) can be accurately mimicked by evaluating velocity vector given by Equation (9) using the equilibrium arrays, and following the rules of motion given by Equations (14–17).

The memristor is a nano-scale passive variable resistor with memory whose resistance changes depending on the polarity and magnitude of a voltage applied to the device terminals and the duration of this voltage application. Threshold condition is one of the key characteristics of the memristor. Based on this feature, small voltages across the memristor, below its threshold (_{th}), do not cause a considerable memristance change, while larger voltages, greater than the threshold, induce much greater memristance changes (Jo et al.,

The emerging technology of memristive circuits is one of the most promising technologies for hardware implementation of computational systems. Accordingly, developing new analog computation approaches compatible with a hybrid memristor-crossbar/CMOS platform is a significant ongoing research area (Bichler et al.,

The proposed memristive circuit is shown in Figure

In the proposed circuit,

Clearly in this representation, increasing or decreasing one unit in the

These units receive the one-hot value of the discrete variables from XR and YR and produce their proportional analog values in the outputs Xa and Ya. The circuit diagram of these units contains an opamp-based multi-input analog adder where the input resistors are replaced by the memristors, so the gain of each input can be controlled by changing the value of its input memristance. Thus, as the Figure

where _{f} is the feedback resistor of the op-amps, _{d} is the voltage level from digital registers representing logic “one." In this circuit, reading conductance of the memristors ought not to affect their value. On the other hand, it is mentioned that if the applied voltage to the memristor is below its threshold (_{th}), it does not induce considerable change in the memristance. Hence, as an essential condition, the applied voltage to the memristors ought to be lower than the threshold value (_{d} < _{th}). The conductance values functions _{mx}(·) and _{my}(·) for the memristors are described as:

where _{x}, _{y} are the normalizing coefficients and _{mx0}, _{my0} are the initial conductance parameters. For uniformity, the output voltage range for both units ought to be clamped in

and for satisfying upper boundary condition, we have:

The parameters can get various values satisfying the conditions described in Equations (22) and (23). Note that there is an undesirable negative sign multiplied in the output voltages Xa and Ya in Equation (20), but it is not the source of any difficulties in our approach, because what is calculated in the next blocks is just the difference of the voltages and the negative sign can be easily handled.

In these units, the equilibrium arrays given by Equation (7) are stored on the memristors, and the analog output voltages proportional to the equilibrium values for current YR state is produced in the _{eqa} and _{eqa} outputs. Thus, the governing equation of these circuits can be given by:

where _{f} is the feedback resistor of the op-amps, _{d} is the voltage level from digital registers representing logic “one,” and the conductance values functions _{eqx}(·) and _{eqy}(·) for the memristors are described as:

Similar to the equations of the X_DAC, Y_DAC units presented in Equation (20), there is an undesirable negative sign in the expression for the output voltages _{eqa} and _{eqa} in Equation (24), but it is not the source of any difficulties. This is because voltage differences are calculated in the next blocks and the sign can be easily handled.

As shown in Figure _{eqa} or _{eqa}) and the _{dx} or _{dy}) using Equation (9). The governing equations of the subtractor circuits are:

where _{sx} and _{sy} are the gain of subtractors. Note that the coefficients α and β in Equation (9), and the aforementioned undesirable negative sign from the previous steps ought to be considered in the gain of the subtractors in this step. In the next step, the adders add the proportional analog voltages of the

For adjusting the mathematical parameters

where _{b} and _{c} are the adjusting parameters. We will determine the essential conditions on how all the introduced parameters in X_vel and V_vel units can be chosen, after introducing the VCO units and their parameters.

These units contain VCOs and controller circuit. The VCO receives the analog voltage corresponding to the velocity (_{x} or _{y}) and produces the linearly proportional motion clock pulse, which is used for clocking the shift of the XR and YR registers, with a specific range of the frequency. Functional equation of the VCOs is given by:

where _{vco} is the VCO coefficient, the UT and LT are, respectively, the upper-hand and lower-hand input voltage threshold of the VCOs and _{max} and _{min} are, respectively, the maximum and minimum output frequencies of the VCOs. The VCOs produce two asynchronous clock pulse signals based on the derived equation on their output pins X_clk and Y_clk. Note that the VCOs inherently satisfy Equations (14) and (15).

The controller circuit determines the direction of the motion (shift direction of the XR and YR state registers) on the pins X_SD and Y_SD. Zero logical value on these pins means that the next change in their proportional state registers is a backward shift, and the one logical value means that the next change is a forward shift:

Eventually, the state registers are shifted at the positive edge of their proportional clock pulse signals X_clk and Y_clk:

where

As we mentioned before, we ought to draw the essential conditions on the coefficients for proper operation of the system. According to the Equation (2), the analog velocity of the state point in

By dividing these two equations, we obtain:

Now, assume the special condition where the vertical distance of the state point from the equilibrium curve is equal to zero and

By dividing these two equations, we have:

Therefore, the parameters must satisfy the essential conditions given by Equations (35–38).

Auxiliary Function: This unit is a logical block for implementing the auxiliary functions like threshold conditions and resetting functions.

The proposed circuit is significantly more efficient in term of area in comparison with the previous similar approach named CMDS (Bavandpour et al.,

Memristor | 400 | 20,000 |

Switch | – | 400 |

Analog adder | 1 | 1 |

VCO | 2 | 2 |

Auxiliary function | 1 | 1 |

State variable registers | 2 | 2 |

Analog output | Yes | No |

One-hot digital output | Yes | Yes |

The behavior of a biological neuron depends on many intrinsic and extrinsic factors, such as the morphology of its dendritic tree, the type and characteristics of ion-gated channels and voltage expressed by the neuron, the location of the stimulating input, and so on. These factors are indeed important, because they determine not only the neuronal response but also the rules that govern dynamics of the neuron. Accordingly, different neuron models are developed to mimic different responses of the neuron and their related dynamics (bifurcations). Our generalized circuit can mimic the dynamics and the various responses of a wide range of neuron models with different computational complexity. In this section, we investigate three main neuron models as the FHN (Fitzhugh,

The FHN is a two-dimensional neuron model derived form the simplified Hodgkin-Huxley (HH) model for biological process of spike generation in squid large axons (Fitzhugh,

where

Our FHN-MDS approach can exhibit all significant qualitative phenomena of the original FHN model and their underlying bifurcations. In this study, four main bio-inspired phenomena of FHN model are individually investigated on our approach.

The original FHN model is capable of producing the absence of all-or-none spikes as it is produced in HH model in response to stimulus current

According to this response, the neuron stops repetitive firing and goes back to a stable resting state as the amplitude of the input current increases. This type of response is based on a special bifurcation scenario in the phase plane representation. When input current

Another basic behavior of the original FHN model is post-inhibitory rebound spikes. This phenomenon is also called anodal break excitation. This behavior is produced in response to application of a short negative pulse to the model. As the negative pulse is applied, hyperpolarization is occurred, and the resting state slides to the left. At the moment when negative pulse is finished, anodal break is occurred, stable equilibrium point promptly shifts up, and the state point makes a transitory large-amplitude excursion to move from previous location of the stable point to its current location and rest. This transient state results in a single spike in time domain. This response is investigated on the FHN-MDS circuit in Figure

This type of response is a common basic dynamical mechanism produced by HH-family models. According to this response, slow increase of the injected current

The AdEx is a two-dimensional neuron model that mathematically describes the dynamical relationship between the membrane potential of the neuron

where _{L} is total leak conductance, _{L} is effective rest potential, Δ_{T} is threshold slope factor, _{T} is effective threshold potential, _{w} is time constant of the adaptation current, _{r} is reset voltage, and _{th}), the exponential term in the equation results in a rapid increase of the membrane potential. The downward portion of the spike shape is produced by an auxiliary reset condition. Subthreshold and spike-triggered adaptations are, respectively, considered using the parameters

According to the explanations in the previous section, the model can be rewritten in the general form of Equation (2) and then mapped easily on the proposed platform. The general form of AdEx neuron model in our mapping is given by:

Despite the simplicity of this two equation model with only a few parameters, this model can reproduce a wide range of physiological firing patterns. Here, we investigate a number of its main responses.

In this response, when a step current is injected into the neuron, it starts to fire repetitively, and produces a spike train with a constant frequency in its output. Figure

In this response, when a step current is injected into the neuron, the neuron starts to fire repetitively and adapt the spike frequency from a relatively high initial frequency to a specific lower frequency. Figure

In this response, when a step current is injected into the neuron, the neuron produces an initial burst of spikes and then starts to fire repetitively with a constant frequency. Figure

In this response, when a step current is injected into the neuron, the neuron produces a repetitive burst of spikes with a constant frequency (first burst may have different shape due to the initial state point). Figure

In this response, when a step current is injected into the neuron, the neuron produces an initial delay by a slow increase in the membrane potential, and then produces a spike train. The frequency of spike train increases by time as a transient phase (accelerating transient) and then the frequency is fixed. Figure

Similar to the delayed accelerating, in this response, when a step current is injected into the neuron, the neuron produces an initial delay by a slow increase in the membrane potential, and then produces a repetitive burst of spikes with a constant frequency. Figure

In this response, when a step current is injected into the neuron, the neuron produces one transient spike and then remained in the resting state. Figure

The AdEx model can represent the irregular spiking behavior, which is a chaotic response, despite this fact that the equations are deterministic. According to this response, inter-spike intervals vary over time without a periodic pattern. Figure

Izhikevich model (Izhikevich,

where the main dynamical variables of the model, ^{+} and Na^{+} channels, and provides controlling feedback to _{th} = 30 mV) and neuron fires. This model can be conveniently rewritten in the general form of Equation (2) and then mapped easily on the proposed platform using the procedure explained in the previous sections. The general form of Izhikevich neuron model in our mapping is given by:

Time-domain neuron-like responses of the proposed 64-bit memristive cellular Izhikevich-MDS approach are shown in Figure

In this section we introduce a potentiometers-based board simulating our memristive system. This board is remarkably useful for improving the level of understanding about MDS approach, and the concept of learning in this approach for devising compatible learning algorithms, and also developing novel dynamical models with the capability of mimicking new responses. Note that detailed fabrication of this approach on the memristor-crossbar/CMOS platform considering all platform constraints is not the main aim of this paper and may be investigated in the future.

For circuit-level realization of the approach, the essential information about the target dynamical system is mapped into arrays of resistors with different resistances and also a number of logic gates for auxiliary functions such as reset equations. In our general circuit, type of the response and the dynamical system can be changed by changing these two parts of the circuit. Hence, we use the programmable resistors known as multiturn potentiometers and the programmable digital logic gates known as FPGA to develop a fully reconfigurable and generalized hardware.

The final board for a 20 × 20 cellular approach is depicted in Figure _{f} = 10 kΩ to implement X_DAC, Y_DAC, X_eqa, Y_eqa units of the circuit shown in Figure _{mx0}, _{my0}, _{x}, _{y} based on our board voltage level and other constraints such as resistance values before we go through the detailed circuit design. In this case, knowing _{f} = 10 kΩ, _{d} = 3.3 V, and the resistance range of the potentiometers, we decided to limit the range of variable resistances to (10–80 kΩ). Therefore, we can easily evaluate the above-mentioned parameters in Equations (21) and (25) by checking the lower boundary (

The gain of other adders, subtractors, and op-amp based absolute value circuits are equal to unit. Also the sign of the velocities are calculated using single-supply TLC272 opamps working in the saturation mode, and comparing the velocity voltage values with the ground voltage. The final absolute values are converted to 8-bit digital values using two built-in ADC of ATmega16 microcontroller, and sent to the Spartan3 XC3S400-4PQ208C FPGA. The VCO units, the state registers, and the auxiliary reset equations are synthesized on the FPGA, and 2 × 20 wires of state registers are connected back from the FPGA to the input of potentiometer arrays.

One of the challenging topics in dynamical systems is error analysis of an approximation of the system such as piece-wise linear approximations (Soleimani et al.,

In the Section 2, in the first step, we changed the representation of the general form of the target dynamical system, and showed that the velocity of a given state point in the phase plane is related to the vertical distance of that point from the nullclines. This step was just a change in the representation form that did not cause any kind of error in the system. Therefore, the cellular mapping, and different variations in fabrication process are the source of error in our system. The detailed circuit implementation of different blocks of the system and investigation of the hardware variations in the system is not in the scope of this paper, and will be studied as a future work. So, here we investigate the effect of cellular mapping.

As explained before, although our approach turns the continuous phase plane into a plane with discrete crossbar cells, it can exactly locate any point in the plane and indicate that point using the motion times. It implies that our asynchronous approach can track any trajectory in the plane with the sufficient number of cells. Figure _{x}, _{y}) for each cell are represented. These times are updated in each cell based on Equations (9) and (11). As shown, the virtual cellular trajectory (cellular trajectory + motion times) can exactly track the continuous trajectory in the phase plane under specific conditions on the trajectory and the number of cells.

Note that the Figure

Here, we separate spike timing error caused by the piece-wise constant velocity nature, and spike shape error caused by cellular nature of our approach. Table

FHN | Tonic spiking | 1.78 | 1.04 | 0.67 | 0.43 | 0.26 |

AdEx | Tonic spiking | 2.29 | 1.34 | 1.00 | 0.79 | 0.54 |

Regular bursting | 3.52 | 1.73 | 1.08 | 0.81 | 0.65 | |

Izhikevich | Tonic spiking | 2.03 | 1.22 | 0.88 | 0.54 | 0.32 |

Regular bursting | 3.01 | 1.69 | 1.01 | 0.76 | 0.55 | |

FHN | Tonic spiking | 3.24 | 1.78 | 1.22 | 0.88 | 0.62 |

AdEx | Tonic spiking | 9.41 | 5.09 | 3.99 | 2.98 | 2.07 |

Regular bursting | 17.55 | 8.77 | 5.04 | 4.57 | 3.95 | |

Izhikevich | Tonic spiking | 7.85 | 4.08 | 3.12 | 2.01 | 1.44 |

Regular bursting | 10.14 | 5.00 | 3.85 | 2.97 | 2.45 |

This paper presented a novel, efficient, fully reconfigurable approach for implementing neuromorphic dynamical systems. This approach shows a huge capability in different aspects, and it can be developed and fabricated for different applications. In this section, we present two main capabilities of our approach and attempt to point out an acceptable roadmap for research works toward which we can obtain a powerful hardware for neuromorphic applications. These two capabilities, which are the key topics in neuromorphic engineering, are (1) configuring and networking, and (2) learning.

Considering various applications of the neural systems, it can be easily concluded that any proposed approach for hardware implementation of a single neural element ought to provide the capability of networking with variable weights. Here, we draw a roadmap to achieve a reconfigurable network of Memristive Dynamical System (MDS) cells with inherent plasticity. As explained, MDS approach produces both analog and digital one-hot outputs, and accepts analog voltage as injected input to the system. This feature makes it feasible to replace conventional hardware implementation of single neural cells with the MDS cells in a wide range of applied digital/analog weighted networking schemes with plasticity to achieve a network of optimum accurate general neural cells. Besides, it is convenient to propose novel networking schemes to connect Memristive Dynamical System (MDS) cells with variable weights, so the resultant system is flexible network of general neural cells with variable intera-cell dynamics and the capability of applying bio-inspired learning schemes.

Figure _{min}, _{max}) and [_{min}, _{max}) on the dynamical variables _{eqx} and _{eqy} and their equivalent memristances (where

Applying the flexible memristors in a well-mapped general hardware brings a significant advantage of learning capability to the MDS approach. In other words, the MDS is capable of learning various intra-cell dynamics with various complexities to reproduce various signal shapes specifically spike patters in its output. One of the major advantages of our novel MDS approach over previous CMDS (Bavandpour et al.,

This study presented an unconventional computing approach based on a novel general mapping for dynamical systems in two-dimensional cellular phase space, and then its hardware implementation on efficient hybrid memristors-crossbar/CMOS memristive circuit. The proposed approach calculates the velocity vector using the vertical distance of the state point from the nullclines, and applies an unconventional VCO-based asynchronous technique to track the state trajectory in the phase space. This approach employed 4^{2} memristors and 2

It is general and highly programmable.

It achieves a relatively high accuracy in a low resolution cellular space.

Its implementation cost is almost independent from the computational effort of the target mathematical model.

It is capable of implementing mathematically indescribable dynamical systems.

It is implementable on a nanoscale efficient memristor/CMOS hardware platform.

It is conveniently feasible to apply conventional analog or digital networking schemes to the MDS cells, and also propose novel networking schemes.

It is capable of learning unknown intra-cell dynamics.

Toward the future roadmap, the problems which ought to be solved can be listed as:

A detailed circuit design and analysis process to achieve an optimum circuit for different blocks of the MDS system.

Developing unconventional learning methods for MDS-based dynamical systems.

A novel compatible mathematical method for stability analysis of the proposed cellular mapping.

This work was partially funded by EU HBP project under grant number FP7-ICT-2013-FET-F-604102, by EU H2020 ECOMODE project under grant agreement 604102, and by Spanish research grant (with support from the European Regional Development Fund) TEC2012-37868-C04-01 (BIOSENSE).

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.