Improvement of the Izhikevich model based on the rat basolateral amygdala and hippocampus neurons, and recognition of their possible firing patterns

Introduction: Identifying the potential firing patterns following by different brain regions under normal and abnormal conditions increases our understanding of what is happening in the level of neural interactions in the brain. On the other hand, it is important to be capable of modeling the potential neural activities, in order to build precise artificial neural networks. The Izhikevich model is one of the simple biologically plausible models that is capable of capturing the most known firing patterns of neurons. This property makes the model efficient in simulating large-scale networks of neurons. Improving the Izhikevich model for adapting with the neuronal activity of rat brain with great accuracy would make the model effective for future neural network implementations. Methods: Data sampling from two brain regions, the HIP and BLA, is performed by extracellular recordings of male Wistar rats and spike sorting is done using Plexon offline sorter. Further data analyses are done through NeuroExplorer and MATLAB software. In order to optimize the Izhikevich model parameters, the genetic algorithm is used. Results: In the present study, the possible firing patterns of the real single neurons of the HIP and BLA are identified. Additionally, improvement of the Izhikevich model is achieved. As a result, the real neuronal spiking pattern of these regions neurons, and the corresponding cases of the Izhikevich neuron spiking pattern are adjusted with great accuracy. Conclusion: This study is conducted to elevate our knowledge of neural interactions in different structures of the brain and accelerate the quality of future large scale neural networks simulations, as well as reducing the modeling complexity. This aim is achievable by performing the improved Izhikevich model, and inserting only the plausible firing patterns and eliminating unrealistic ones, as the results of this study.


Introduction
One of the pivotal components of the brain's microscopic structure is a neuron cell. The importance of this concept has led to comprehensive researches to understand the mechanism behind neuronal activity. One of the major results is that unlike other body cells, neurons interact with each other by receiving and sending electric pulses or spikes.
Each spike causes the release of neurotransmitters which results in changes in the activity of its neighboring neurons.
Spiking neural networks (SNNs) that are the third neural networks generation (Maass, 1997) have been developed to imitate the natural neural networks. Spiking neural networks follow the same trend as computational neuroscience. The ultimate goal of both is to realistically represent and configure the functionality of different brain areas. SNNs originated from the study of Hodgkin and Huxley (Hodgkin & Huxley, 1952) in 1952.
The fundamental aim of SNNs is to encode all the information related to single spikes rather than just their firing rate (Maass & Bishop, 2001). Spiking  Moreover, spiking neural networks facilitate the understanding of human brain for researchers (Kuebler & Thivierge, 2014). Computational studies of SNN conducted by Maass and Schmitt (Maass, 1997;Maass, 1995a;Schmitt, 1998) have shown great efficiency of the third neural networks generation.
In each dynamical study, one critical issue is which model can describe spiking dynamics of the neuron more efficiently. Although the Hodgkin-Huxley model can greatly simulate the biological functioning of a neuron, it involves 12 equations consisting four differential equations, and three parameters to model the activity of one neuron (Johnson & Chartier, 2017). This complex modeling results in a very expensive implementation. Also, the Hodgkin-Huxley model fails to exhibit the all-or-nothing firing mechanism for action potential generation (Deng, 2017).
A popular model that represents a quite good compromise between computational efficiency and biologically realistic behaviors is the Izhikevich model (Izhikevich, 2003).
This model is not only biologically plausible, similar to Hodgkin-Huxley model in this sense, but also it is as efficient as integrate-and fire model, computationally. The Izhikevich model is also capable of simulating large-scale spiking neurons in real time (Izhikevich, 2004). Therefore, in order to empower the neural interaction modeling based on real data, elevating the accuracy of the Izhikevich model in representing the neurons activity seems prominent. This will also result in increasing the application of Izhikevich model as an efficient model in implementation of functional neural networks. One of the techniques to improve this model and adapt it to the behavior of considering real neurons is to optimize its parameters.
Genetic algorithms have developed according to basic concepts in evolution and imitation of natural processes (Holland, 1975). These criteria consist of mutation, recombination, and assortment of populations in a synthetic environment. The substantial components required in developing genetic algorithms, introduced by Bremermann in techniques consisting genetic algorithms increased the popularity of these algorithms in the following years (Rechenberg, 1973;Schwefel, 1974 identification of single neurons of these structures under brain's normal activity can play an important role in differentiating the normal and abnormal patterns associated with them. Furthermore, the ability to represent the firing activity of these structures' neurons with a mathematical model can conduct to better formulation and simulation of the problem and ultimately, more effective treatment for certain associated diseases. Developing electrophysiological recording of single neurons activity provides a basis for exploring the structure of brain functions. However, the recorded signals are mostly contaminated by a high amount of background noise; noise from the recording system or the activity of distant neurons. Moreover, the recorded data is related to the activity of a number of neurons adjacent to the recording site (Lewicki, 1998). Analysis of neural recordings requires one of the complicated interpretation tools that is known as spike sorting.
Spike sorting is the process of isolating action potentials from the background activity which is considered as noise, extracting prominent spike features from the detected spike waveforms, and finally allocating spikes with same features to their originating neuron (Takekawa, Isomura, & Fukai, 2010;Rutishauser, Schuman, & Mamelak, 2006). This process can be done by an appropriate choice of clustering methods. Then, clusters of spikes can be used for further analysis and modeling.
In this study, one main objective is to identify the possible firing patterns that the neurons of the HIP and BLA follow under normal activity. Another noticeable following goal is to improve the Izhikevich model to make it more accurate in representing the firing activity of rat brain real data.

Experimental Implementation
Male Wistar rats were used to investigate neuronal electric signaling in the normal BLA and HIP. Each rat was housed in Animal Care Facility maintained at 23 ± 1○C on a

Electrophysiological recording and data collection
Animals' anesthesia was achieved using urethane with initial dose of 1.5 g/kg, intraperitoneally. Additional doses were given whenever needed to maintain surgical anesthesia depth as checked by foot pinch and corneal reflex. To remove the potential pain, 0.1 ml buprenorphine was injected, subcutaneously. Conducting tracheotomy, rats were located in a stereotaxic instrument. Using a heating pad, the rat body temperature was maintained for the experiment duration. Electrophysiological recordings of the firing activity of neurons in the HIP and BLA were performed via an acute microelectrode with one channel. Each channel records the electric activity of a few neurons adjacent to it; the activity of farther neurons appeared as the background noise due to their low amplitude.
The microelectrode was proceeded to the left BLA (AP: -2.52 mm and ML: -4.8 mm from the bregma, and DV: -8.4 mm from the surface of skull) and the left HIP (AP: -3 mm and ML: -1.8 mm from the bregma, and DV: -3 mm from the skull surface) according to the rat brain atlas (Paxinos & Watson, 2007). Signals were recorded using a data acquisition system, filtered between 300 and 10000 Hz, and sampled with the rate of 50 kHz. Each recording lasted for 1800 seconds.

Data Analysis
Recorded data from the electric activity of neurons were exported to and analyzed via an offline sorter software called Plexon (Plexon Inc., Dallas, TX). Spikes were detected through manual amplitude threshold discrimination. The threshold level discerns a trade-off between the missed spikes and the noise which may pass that level. The Auto-correlogram displays a single spike train against itself. Another tool that compares arrival times of spike trains is cross-correlogram. Through cross-correlogram, different identified clusters of spikes were explored to validate the exact number of neurons in each set of recorded data. Finally, the average firing rate histograms were generated and verified for all neurons, over the entire period of 1800 seconds. Then, validated clusters of spikes were exported to MATLAB software to be used for modeling. This software is also used to code our desirable genetic algorithm, Izhikevich model and depict the comparison figures of different firing patterns.

Izhikevich neuronal model
The two dimensional Izhikevich neuronal model (Izhikevich, 2003) is defined by three equations as follows: Where variables and are the membrane potential of the neuron and membrane recovery variable, respectively. Activation of + ionic currents and inactivation of + ionic currents can be represented by the variable . This variable supplies with a negative feedback. Variable represents the delivery of synaptic currents. Equation (3) activates when the amplitude of action potential reaches the threshold +30 . , , , and are dimensionless parameters of the model. based on their response to the applied dc current (Izhikevich, 2010). The BLA and HIP neurons that have been used in this study follow neuronal behaviors similar to some of these neuro-computational properties. The following properties are tonic spiking (Nessler & Bernhard, 2013), phasic spiking (Malsburg, 1999), mixed model (Connors & Gutnick, 1990), integrator, rebound spike, threshold variability (Izhikevich, 2003), depolarizing after-potentials (Malsburg, 1999), and inhibition-induced spiking (Izhikevich, 2003).

Genetic Algorithm
The genetic algorithm is one of the well-known evolutionary algorithms that employs the principle of best populations' selection in each iteration for the whole process. This property provides the opportunity to select and generate individuals that are more adapted to the environment and remove the ones with less consistency. By repeating the same process for several generations and replacing undesirable populations by more adjustable ones, the algorithm evolves a population with optimal characteristics. • Replacement: in this step, next generation produces by replacing the primary solutions or some parts of them with the new desirable ones which are generated via crossover and mutation.

Results
As mentioned before, the parameters of the Izhikevich model have different values to exhibit different potential firing patterns of neurons. In this study, we optimized each set of parameters by modifying our optimization problem variables such as maximum number of iterations, crossover percentage, mutation rate, etc., in the performed genetic algorithm to minimize the associated error.
In this paper, recording data from two regions of the rat brain consists of the BLA and HIP under normal activity, spike sorting is done via Plexon Offline Sorter. The process resulted in three clusters of spikes for each region. Based on spike sorting criteria, each cluster represents the activity of one single neuron adjacent to the recording site.
Afterward, we compared the firing patterns of the original Izhikevich model, model with optimized parameters , , , and , and the firing behavior of the mentioned regions real single neurons.

Parameters optimization and neural firing results
The main step after applying spike sorting was to recognize the firing patterns that each single neuron of the data was following. This way, we traced the activity of each neuron in a specific period of time and compared them with the known firing patterns.
Next step was to design a proper genetic algorithm to optimize corresponding cases of the Izhikevich model parameters. As mentioned before, in designing the proper genetic algorithm, the values of optimization problem variables were dependent on different cases of the Izhikevich neural pattern and data. So, in order to reach a customize genetic algorithm for each case, different tests were run with different variable amounts. The fitness criterion was the error minimization (mean square error) of the neural action potential difference between the Izhikevich and real neurons. As a case example, considering the designed genetic algorithm for tonic spiking pattern for the BLA neurons, the optimal crossover and mutation ratio are assigned 0.7 and 0.8, respectively. The algorithm terminated in 150 generations.
In this paper, we represent the results according to the potential firing patterns Moreover, results represent that first cluster may follow a firing pattern of each of improved integrator, phasic spiking, depolarizing, rebound spiking, or threshold variability, as depicted in Fig. 1. The second and third clusters followed a firing pattern of the improved Izhikevich pattern for inhibition-induced spiking and tonic spiking, respectively. The activity pattern of these two clusters is depicted in Fig. 2 (a) and (b).
In Fig. 2 (c) to (e), we showed three neural behaviors that a real neuron of the BLA did not follow and compared them with either firing pattern of Izhikevich neuron or Izhikevich improved neuron. Mentioned firing behaviors consisted of mixed spiking, bistability, and spike frequency adaptation.

The HIP
Similar to the previous discussed region of the rat brain, the HIP neurons have greatly followed the improved Izhikevich pattern for each of the sorted clusters. The results can be seen in Fig. 3 (a) to (d). For all considered rats, the three single neurons extracted from data recording of this region had the firing pattern as follows: one cluster followed improved mixed or tonic spiking, the second cluster followed the improved inhibition-induced spiking, and the last one followed improved tonic spiking. Fig. 3 (e) and (f) represented two firing activity which the HIP single neurons may not follow.
These patterns are bistability and spike frequency adaptation. As mentioned earlier, the Izhikevich model is capable of representing the firing pattern of most known types of cortical neurons according to changes in its parameters values. However, it fails to represent the neuronal firing pattern of some specified parts of rat cortex such as the HIP and BLA with great accuracy. Shown by the real data in this study, one potential problem of the Izhikevich model is its after spike potential; the Izhikevich neuron potential returns to the amount of parameter , as it is shown in supplementary Fig. 1 (Izhikevich, 2003). Nevertheless, considered cases in current exploration did not follow this potential reset. In Fact, they simply returned to their initial amount, as it can be seen in Fig. (1) to ( Sophisticated real-world problems and attempts to find appropriate solutions for them have led scientists to investigate natural phenomena and imitate them for years.

Discussion
Optimization algorithms have been developed based on the natural processes progressively in past decades (Michalewiez, 1996). Some outstanding algorithms such as evolutionary algorithms and the genetic algorithms perform intelligent searches in the massive space of solutions considering required statistical techniques. Natural approach followed by these algorithms results in achieving optimal solutions for natural phenomena such as spiking activity of neurons. This way, one of the best optimization algorithms, genetic algorithm is used in this inquiry. To strengthen the validity of the achieved results, data recording has performed for several rats under anesthesia, and the whole analysis processes repeated for acquired data. The firing activity of all BLA and HIP neurons has compared; for the recorded data from each region, results were predominantly consistent.  The second cluster may follow inhibition induced spiking and the third cluster may track tonic spiking. Plots (c) to (e) illustrate three possible firing patterns that may not be followed by the BLA neurons. In these plots, the comparison of second and third cluster firing pattern of the BLA, mixed spiking, bistability, and spike frequency adaptation from Izhikevich firing pattern, and improved ones are depicted. Figure 3: Comparison of the HIP neurons firing patterns. Figure 3 (a) to (d) shows the comparison of the possible firing patterns of the rat HIP that are tonic spiking, mixed spiking, and inhibition induced spiking mode from Izhikevich firing pattern, and improved patterns. Figure 3 (e) and (f) represents unfollowed firing patterns that are bistability for the first cluster and spike frequency adaptation for the third cluster of the HIP neuron.