Chaotic Firing Research Articles

A Hardware Chaotic Neural Network With Gap Junction Models

ABSTRACTWe aim at the engineering applications of reservoir computing using hardware chaotic neural networks, including associative memory recall. The reservoir layer used in reservoir computing is networked and constructed using pulse‐type hardware chaos neuron models (P‐HCNMs). The structure of the reservoir layer is simple, which is advantageous for hardware implementation. By inducing chaos in the reservoir layer, it is possible to use the “chaotic edge” where the reservoir reaches its highest efficiency. It has also been reported that incorporating self‐correction within the reservoir layer increases the efficiency of the task. In this paper, we constructed a hardware small‐world neural network using a synaptic model with spike timing‐dependent synaptic plasticity (STDP) and a gap junction model. As a result, it is clarified that all cell body models with synaptic model connections show chaotic firing by simulation at the same time, and that the STDP model enables learning while keeping the chaotic phenomena. In addition, comparison with the firing of cell body models coupled only with synaptic models suggested that the gap junction model works significantly in inducing chaos in neural networks.

Effect of autaptic synapse on signal transmission performance of dressed Hodgkin–Huxley neuron

Astrocytes, which have complex relationships with neurons, are the most important biological arguments that support neurons in the performance of cognitive functions. Another important biological argument in nervous systems is autapse, a special type of synapse through which the neuron provides feedback to itself. It is well known that both astrocyte and autapse modulate neuron dynamics and they affect memory functions. Considering the idea that some cognitive brain functions occur through chaotic firings, more complex models in which structures such as astrocytes and autaptic synapses are included in the system are needed to better understand the neural system. In this study, the signal detection abilities of astrocyte-dressed Hodgkin–Huxley neurons which have electrical, excitatory, and inhibitory chemical autaptic synapses are examined separately under chaotic environmental conditions. The results show that astrocyte-supported neurons, especially those with electrical and excitatory-chemical autapse, exhibit a strong chaotic resonance for appropriate parameters. After the autaptic synapse, the astrocyte is a second modulator of the neuron’s signal detection success. Additionally, it is observed that autaptic time delay is more effective than autaptic conductance in modulating the signal transmission success of the neuron. For all autapse types, the parameter values at which the supportive effect of the astrocyte in the model is obtained are presented. Furthermore, multiple chaotic resonance is detected in neurons with electrical and excitatory chemical autapse when the autaptic delay is equal to integer multiples of the weak signal period. This is observed for neurons with inhibitory chemical autapse when half of the weak signal period is added to whole multiples of the weak signal period. We hope that our findings will shed light on the chaotic environment behavior of complex nervous systems interacting with astrocytes and autapse.

Dynamic Effects Analysis in Fractional Memristor-Based Rulkov Neuron Model.

Mathematical models such as Fitzhugh-Nagoma and Hodgkin-Huxley models have been used to understand complex nervous systems. Still, due to their complexity, these models have made it challenging to analyze neural function. The discrete Rulkov model allows the analysis of neural function to facilitate the investigation of neuronal dynamics or others. This paper introduces a fractional memristor Rulkov neuron model and analyzes its dynamic effects, investigating how to improve neuron models by combining discrete memristors and fractional derivatives. These improvements include the more accurate generation of heritable properties compared to full-order models, the treatment of dynamic firing activity at multiple time scales for a single neuron, and the better performance of firing frequency responses in fractional designs compared to integer models. Initially, we combined a Rulkov neuron model with a memristor and evaluated all system parameters using bifurcation diagrams and the 0-1 chaos test. Subsequently, we applied a discrete fractional-order approach to the Rulkov memristor map. We investigated the impact of all parameters and the fractional order on the model and observed that the system exhibited various behaviors, including tonic firing, periodic firing, and chaotic firing. We also found that the more I tend towards the correct order, the more chaotic modes in the range of parameters. Following this, we coupled the proposed model with a similar one and assessed how the fractional order influences synchronization. Our results demonstrated that the fractional order significantly improves synchronization. The results of this research emphasize that the combination of memristor and discrete neurons provides an effective tool for modeling and estimating biophysical effects in neurons and artificial neural networks.

A gas sensing neural circuit for an olfactory neuron

A gas sensor can convert external gas concentration or species into electric voltage or current signals by physical adsorption or chemical changes. As a result, a gas sensor in a nonlinear circuit can be used as a sensitive sensor for detecting external gas signals from the olfactory system. In this paper, a gas sensor and a field-effect transistor are incorporated into a simple FithzHugh–Nagumo neural circuit for capturing and encoding external gas signals. An improved functional neural circuit is obtained, and the effect of gas concentration, gas species and neuronal activity can be discerned as the gate voltage, threshold voltage and activation coefficient of the field-effect transistor, respectively. The gas concentration can affect the neural activities from quiescent to normal working and, finally, to saturation state in bursting, spiking, periodic and chaotic firings with different frequencies. The effects of gas species and neuronal activity on the firing state can also be achieved in this functional neural circuit. In addition, variations in the gate voltage, threshold voltage and activation coefficient can cause switching between different firing modes. These results can be helpful in designing artificial olfactory devices for bionic gas recognition and other coupled systems arising in applied sciences.

Further study on the Rulkov neuron: geometric control, circuit realization and chaos-based encryption

Rulkov neuron with various firing modes is exhaustively explored by numerical simulation aiming to disclose its unique properties of evolution. Transient chaos and the coexistence of periodic state and chaos are found. Furthermore, it is found that the chaotic attractors in the neuron can also be arranged flexibly in phase space by a single offset booster, while the amplitude of firings is rescaled by an independent amplitude controller. Circuit implementation based on MCU is carried out demonstrating the predicted dynamics. Confidentiality and security of data play an important role in safeguarding national production and livelihood. In this paper, the chaotic firing is introduced into the Tiny Encryption Algorithm (TEA) for image encryption based on MCU combined with metrics analysis. It is proven that the Chaos-based Tiny Encryption Algorithm (ChaosTEA) exhibits higher efficiency and security compared to the traditional TEA algorithm.

A joint image encryption based on a memristive Rulkov neuron with controllable multistability and compressive sensing

Image encryption, as a critical branch, has attracted increasing attention to the demand for information security specifically in the era of artificial intelligence (AI). A chaotic sequence is regarded as an important encryption source, and compressive sensing provides an effective technology for obtaining and reconstructing sparse or compressible signals in applied electronic engineering. In this work, the detouring matching pursuit algorithm and DNA coding are utilized to increase the performance based on a newly developed chaotic firing neuron. A memristor as the electromagnetic component is proven to enhance synaptic plasticity and emulate the synaptic connections in the brain. A unique discrete memristive neuron is derived for exploring the dynamics of neuron firing. By modifying the feedback from the memristor various coexisting neuronal chaotic firing are possessed. Because of the periodic evolution of the resistor from the memristor, the derived memristive Rulkov neuron exhibits coexisting homogeneous and heterogeneous multistability, which enables amplitude controllability and different types of coexisting chaotic firings. Circuit implementation based on CH32 is built to verify the controllability of the coexisting dynamics.

Ion channels and the diversity of spontaneous firing in anterior pituitary corticotrophs: A dynamical analysis

Pituitary cells release hormones based on firing patterns, mediated by multiple ionic currents. Analyzing a single channel’s effect on firing is insufficient. We used a four-dimensional Hodgkin–Huxley-type corticotroph model to analyze its dynamics from a geometric perspective. We conducted a two-parameter bifurcation analysis of the system and found that activation levels of BK, NS, Kir, Ca, and LCa channels are crucial in generating and transitioning between different firing patterns. Chaotic firing can arise during the transition between firing patterns. We have employed the Poincaré map method and sequence reconstructions to construct return maps to characterize chaos-firing. The membrane potential time series is affected by fluctuations in intracellular calcium ([Ca2+]i). We studied the variable c using fast–slow analysis techniques and phase portraits to explore its relationship with phase portraits. The study revealed that the system exhibits bistability, which is dependent on the initial values of c or V. The findings indicate that firing activities can arise from either a supercritical or subcritical Hopf bifurcation. Analyses show the interaction between the membrane potential and [Ca2+]i at various BK channel activation levels.

Intrinsic mechanism of influence of channel blocking and noise on response state of multicompartmental neurons

The fine structure of multi-compartment neurons can simultaneously capture both temporal and spatial characteristics, offering rich responses and intrinsic mechanisms. However, current studies of the effects of channel blockage and noise on neuronal response states are mainly limited to single-compartment neurons. This study introduces an analytical method to explore theintrinsic mechanism of channel blockage and noise effects on the response states of multi-compartment neurons, by using the smooth Pinsky-Rinzel two-compartment neuron model as a case study. Potassium, sodium, and calcium ion channel blockage coefficient are separately introduced to develop a smooth Pinsky-Rinzel neuron model with ion channel blockage. Methods such as single-parameter bifurcation analysis, double-parameter bifurcation analysis, coefficient of variation, and frequency characteristics analysis are utilized to examine the effects of various ion channel blockages on neuronal response states. Additionally, smooth Pinsky-Rinzel neuron Subunit noise model and conductance noise model are constructed to investigate their response characteristics by using interspike interval analysis and coefficient of variation indicators. Subthreshold stimulation is used to explore the presence of stochastic resonance phenomena. Single-parameter bifurcation analysis of the ion channel blockage model elucidates the dynamic processes of two torus bifurcations and limit point bifurcations in Pinsky-Rinzel neuron firing under potassium ion blocking. Double-parameter bifurcation analysis reveals a nearly linear increase in the Hopf bifurcation node of potassium ions with input current, whereas sodium ions exhibit a two-stage pattern of linear decline followed by exponential rise. The analysis of average firing frequency and coefficient of variation indicates that the moderate potassium channel blockage promotes firing, sodium channel blockage inhibits firing, and calcium channel blockage shows the complex characteristics but mainly promotes firing. Subthreshold stimulation of the channel noise model demonstrates the stochastic resonance phenomena in both models, accompanied by more intense chaotic firing, highlighting the positive role of noise in neural signal transmission. The interspike interval and coefficient of variation indicators show consistent variation levels for both noise models, with the conductance model displaying greater sensitivity to membrane area and stronger encoding capabilities. This study analyzes the general frequency characteristics of potassium and sodium ions in a multi-compartment neuron model through ion channel blocking model, providing special insights into the unique role of calcium ions. Further, the study explores stochastic resonance by using ion channel noise model, supporting the theory of noise-enhanced signal processing and offering new perspectives and tools for future studying complex information encoding in neural systems. By constructing an ion channel blockage model, the effects of potassium and sodium ions on the frequency characteristics of multi-compartment neurons are analyzed and the special influences of calcium ions are revealed. Using the ion channel noise model, the stochastic resonance is investigated, supporting the theory that the noise enhances signal processing. This research offers a new perspective and tool for studying the complex information encoding in neural systems.

Initial condition-offset regulating synchronous dynamics and energy diversity in a memristor-coupled network of memristive HR neurons

Memristors can be thought of as special connection synapses with internal states that regulate synchronous behavior or energy propagation between neurons in a neural network. By connecting two memristive Hindmarsh-Rose (mHR) neurons with one memristor coupling, this paper synthesizes a memristor-coupled mHR neuron (MC-mHRN) network without equilibrium point. For this network, the synchronization condition regulated by the initial condition-offset (ICO) of the memristive channel are derived theoretically, and the ICO-regulating synchronous dynamics is evaluated numerically. Furthermore, the Hamiltonian energy of mHR neuron is studied by Helmholtz's theorem, and the ICO-regulating energy diversity is analyzed numerically. It is demonstrated that the coupling strength and initial condition of the memristive channel can regulate the synchronous dynamics and energy diversity in the MC-mHRN network, thus achieving complete synchronization as well as energy balance. Consequently, the two mHR neurons in hidden chaotic firing modes deliver field energy via memristive channel until the fully synchronized energy balance is achieved. In addition, a digital platform of the MC-mHRN network is fabricated, and the experimental results validate the numerical ones of the synchronous dynamics.

Dynamics and synchronization in a memristor-coupled discrete heterogeneous neuron network considering noise

Research on discrete memristor-based neural networks has received much attention. However, current research mainly focuses on memristor–based discrete homogeneous neuron networks, while memristor-coupled discrete heterogeneous neuron networks are rarely reported. In this study, a new four-stable discrete locally active memristor is proposed and its nonvolatile and locally active properties are verified by its power-off plot and DC V–I diagram. Based on two-dimensional (2D) discrete Izhikevich neuron and 2D discrete Chialvo neuron, a heterogeneous discrete neuron network is constructed by using the proposed discrete memristor as a coupling synapse connecting the two heterogeneous neurons. Considering the coupling strength as the control parameter, chaotic firing, periodic firing, and hyperchaotic firing patterns are revealed. In particular, multiple coexisting firing patterns are observed, which are induced by different initial values of the memristor. Phase synchronization between the two heterogeneous neurons is discussed and it is found that they can achieve perfect synchronous at large coupling strength. Furthermore, the effect of Gaussian white noise on synchronization behaviors is also explored. We demonstrate that the presence of noise not only leads to the transition of firing patterns, but also achieves the phase synchronization between two heterogeneous neurons under low coupling strength.

Firing pattern in a memristive Hodgkin–Huxley circuit: Numerical simulation and analog circuit validation

Spiking firing patterns and their hardware implementation can assist us in exploring spike-based applications. The classical Hodgkin–Huxley circuit can generate spiking firing patterns, but it is hard to implement on analog platform because of the involvement of complex exponential nonlinearities for characterizing the sodium and potassium ion channels. To solve this issue, this paper proposes a second-order local active memristor (LAM) and a first-order LAM to respectively characterize the sodium and potassium ion channels in the Hodgkin–Huxley circuit, thereby obtains a memristive Hodgkin–Huxley circuit. Numerical simulations demonstrate that the memristive Hodgkin–Huxley circuit can generate abundant spiking firing patterns, i.e. periodic and chaotic spiking firing patterns. These abundant firing patterns can be effectively regulated by memristor- and stimulus-related parameters. Moreover, a PCB-based analog circuit employing off-the-shelf discrete circuit components is manually constructed and hardware experiments are executed. Experimental results captured from hardware measurements satisfactorily verify the numerically simulated spiking firing patterns and effectively exhibit the availability of the memristive Hodgkin–Huxley circuit in generating spiking firing patterns.

Dynamic analysis of the discrete fractional-order Rulkov neuron map.

Human evolution is carried out by two genetic systems based on DNA and another based on the transmission of information through the functions of the nervous system. In computational neuroscience, mathematical neural models are used to describe the biological function of the brain. Discrete-time neural models have received particular attention due to their simple analysis and low computational costs. From the concept of neuroscience, discrete fractional order neuron models incorporate the memory in a dynamic model. This paper introduces the fractional order discrete Rulkov neuron map. The presented model is analyzed dynamically and also in terms of synchronization ability. First, the Rulkov neuron map is examined in terms of phase plane, bifurcation diagram, and Lyapunov exponent. The biological behaviors of the Rulkov neuron map, such as silence, bursting, and chaotic firing, also exist in its discrete fractional-order version. The bifurcation diagrams of the proposed model are investigated under the effect of the neuron model's parameters and the fractional order. The stability regions of the system are theoretically and numerically obtained, and it is shown that increasing the order of the fractional order decreases the stable areas. Finally, the synchronization behavior of two fractional-order models is investigated. The results represent that the fractional-order systems cannot reach complete synchronization.

Variations of the spontaneous electrical activities of the neuronal networks imposed by the exposure of electromagnetic radiations using computational map-based modeling.

The interaction between neurons in a neuronal network develops spontaneous electrical activities. But the effects of electromagnetic radiation on these activities have not yet been well explored. In this study, a ring of three coupled 1-dimensional Rulkov neurons and the generated electromagnetic field (EMF) are considered to investigate how the spontaneous activities might change regarding the EMF exposure. By employing the bifurcation analysis and time series, a comprehensive view of neuronal behavioral changes due to electromagnetic inductions is provided. The main findings of this study are as follows: 1) When a neuronal network is showing a spontaneous chaotic firing manner (without any external stimuli), a generated magnetic field inhibits this type of behavior. In fact, EMF completely eliminated the chaotic intrinsic behaviors of the neuronal loop. 2) When the network is exhibiting regular period-3 spiking patterns, the generated magnetic field changes its firing pattern to chaotic spiking, which is similar to epileptic seizures. 3) With weak synaptic connections, electromagnetic radiation inhibits and suppresses neuronal activities. 4) If the external magnetic flux has a high amplitude, it can change the shape of the induction current according to its shape 5) when there are weak synaptic connections in the network, a high-frequency external magnetic flux engenders high-frequency fluctuations in the membrane voltages. On the whole, electromagnetic radiation changes the pattern of the spontaneous activities of neuronal networks in the brain according to synaptic strengths and initial states of the neurons.

Firing patterns and synchronization of Morris-Lecar neuron model with memristive autapse

Autapse is a self-feedback current in some neurons that can substantially affect the neuron’s characteristics. Due to the importance of electromagnetic induction in the dynamical functions of the neurons, the memristive autapses have been introduced. This paper presents the 3D Morris-Lecar neuron model with a memristive autapse. The dynamics of the memristive autaptic model are investigated by varying autapse parameters. It is revealed that the memristive autapse can change the period of the neuron’s firing or even change its dynamics between periodic and chaotic firings. The results represent that the chaotic regions are extended with the increase in the autapse time delay. Moreover, the memristive autapse leads to the emergence of multistability. The synchronization of coupled memristive autaptic neurons is also examined. It is found that the synchronization of chaotic neurons is enhanced by considering the memristive autapse. With increasing the autapse gain, the required coupling strength for synchronization decreases.

Response mechanism in a functional neuron under multiple stimuli

Based on an auditory neuron model developed from the piezoelectric neural circuit, two different stimuli (F1, F2) are respectively used and then combined (F1+F2) synchronously to trigger different firing modes (S1, S2, S3). The Hamilton energy for each firing mode is calculated to discern the effect from the external stimulus, and the final firing mode is mainly controlled by one of the stimuli which can induce a higher Hamilton energy in the neuron. The external forcing inducing spiking has priority than the external stimulus inducing bursting to be responded by the neuron. The involvement of periodic firing modes is helpful to induce nonlinear resonance and thus new periodic firing modes can be generated even it presents lower Hamilton energy. The changes in the profile of periodic attractors confirm that external stimuli for inducing spiking and bursting can be perceived sensitively. In case of chaotic firing modes resulting from chaotic driving, the contribution from spiking, bursting and even periodic forcing can be detected because the profile of chaotic attractors is also changed greatly. A correlation function is defined to discern the effect of external stimuli on the selection of firing modes in the neuron driven by multi-channel stimuli. The results confirmed that the neuron is more sensitive to the external stimulus which can release higher Hamilton energy and/or induce high regularity in the neural activities. This study discovers the potential response mechanism in the auditory neuron under external stimuli.

Extreme multistability and phase synchronization in a heterogeneous bi-neuron Rulkov network with memristive electromagnetic induction.

Memristive electromagnetic induction effect has been widely explored in bi-neuron network with homogeneous neurons, but rarely in bi-neuron network with heterogeneous ones. This paper builds a bi-neuron network by coupling heterogeneous Rulkov neurons with memristor and investigates the memristive electromagnetic induction effect. Theoretical analysis discloses that the bi-neuron network possesses a line equilibrium state and its stability depends on the memristor coupling strength and initial condition. That is, the stability of the line equilibrium state has a transition between unstable saddle-focus and stable node-focus via Hopf bifurcation. By employing parameters located in the stable node-focus region, dynamical behaviors related to the memristor coupling strength and initial conditions are revealed by Julia- and MATLAB-based multiple numerical tools. Numerical results demonstrate that the proposed heterogeneous bi-neuron Rulkov network can generate point attractor, period, chaos, chaos crisis, and period-doubling bifurcation. Note that extreme multistability are disclosed with respect to initial conditions of memristor and gated ion concentration. Coexisting infinitely multiple firing patterns of periodic firing patterns with different periodicities and chaotic firing patterns for different memristor initial conditions are demonstrated by phase portrait and time-domain waveform. Besides, the phase synchronization related to the memristor coupling strength and its initial condition is explored, which suggests that the two heterogeneous neurons become phase synchronization with large memristor coupling strength and initial condition. This also reflects that the plasticity of memristor synapse enables adaptive regulation in keeping energy balance between the neurons. What's more, MCU-based hardware experiments are executed to further confirm the numerical simulations.

Synchronization, extreme multistability, and its control for a field coupled neurons with time delays

Neurons are connected through synapses, but it is not reasonable to equate the connection mode of synapses to an edge in the network due to synaptic plasticity whereas the magnetic field coupling can be considered to handle it. Therefore, in this paper, the interaction between HR neurons is realized by magnetic field coupling based on induction coil, and time delay is introduced to represent the lag in information transfer. First, the coexisting firing activities and synchronization behaviors in dual neuronal networks are numerically calculated, respectively, depending on the external stimulation current, coupling strength, time delays, and initial conditions. When the time delays are given, it is interesting to note that the infinite number of firing modes including chaotic firing, periodical firing, and quiescent state is induced by initial conditions. Due to the initial values, the types of synchronization consisting of complete synchronization, delayed synchronization, and asynchronization are then revealed under the framework of extreme multistability. In particular, the state of complete synchronization exhibits only quiescent state and period-1 firing when the time delay is not equal to 0. Furthermore, the linear augmentation method is conceived to control extreme multistability. It can be found that the attractors with different positions and topological structures can be controlled to the point attractors with the same shape but with different positions when the coupling strength of linear system and nonlinear system is increased. That is, the heterogeneous multistability can be successfully controlled to the homogeneous multistability, and the coupled neurons can also be achieved synchronization after control. These conclusions in this paper could be helpful in providing new insights for studying neurodynamics and applying neural circuits.

Effects of memristive autapse under field coupling on transition of collective dynamics in neural hypernetworks

Electrical and chemical synapses play important role in neural system. Recently, a neural hypernetwork by taking a unidirectional chemical synapse and bidirectional electrical synapse has been proposed. Electromagnetic radiation is also important to dynamical response and mode transition in firing activities of neurons. In this paper, we first investigate the effect of memristive autapse on transformation of firing activity modes on an individual Hindmarsh-Rose (HR) neuron. Different periodic firing activity patterns and chaotic firing activity modes are observed, and mode transition between these firing patterns can be induced by adjusting the value of delay time of memristive autapse. Then, we consider a HR neural hypernetwork with memristive autapse under field coupling. Collective behavior is studied numerically, and different collective dynamics are observed. Chaotic states, chimera states, imperfect traveling chimera states, traveling chimera states, synchronous coherent states are found when coupling parameters and delay time of memristive autapse are varied. These findings indicate that memristive autapse under field coupling plays a crucial role in neural networks.

Discrete hybrid Izhikevich neuron model: Nodal and network behaviours considering electromagnetic flux coupling

We analyse the dynamics of the improved discretised version of the well known Izhikevich neuron model under the action of external electromagnetic field. It is found that the improved three dimensional IZH map shows rich dynamics. With the variation of the electromagnetic field, period-doubling route to chaos in a repeating fashion is observed from the bifurcation diagram. Even the forward and backward continuation bifurcation diagram which do not completely overlap suggests that there is multistability in the system. The phenomenon of bistability (coexistence of periodic and chaotic attractors) is observed. The presence of periodic and chaotic attractor is aided by the maximal Lyapunov exponent diagram. The Lyapunov phase diagram of electromagnetic field and synapses current shows a large parameter region of chaotic and periodic behaviours with the presence of unbounded regions as well. The IZH map shows a plethora of spiking and bursting patterns such as mixed-mode patterns, tonic spiking, phasic spiking, steady spikes, regular spikes, spike bursting, periodic bursting, phasic bursting, chaotic firing etc with the variation of electromagnetic coupling strength and the synapses current. We also investigate the presence of chimera states in a ring-star, ring, star networks of IZH map neurons. Chimera states are found in the case of ring-star and ring network while synchronised clusters were found in the case of star network and are aided by the spatiotemporal plots, space-time plot, recurrence plots. The rich dynamics shown by the discretised IZH map makes it a promising research model to study about neurodynamics.

Diverse Dynamic Behaviors and Firing Activities of the Modified Fractional-Order Hindmarsh–Rose Neuronal Model Induced by Fractional-Order

It is important to investigate the firing activities of neurons, and previous experimental works have shown that fractional-order neuronal models depict the firing rate of neurons more verifiably. In this study, a modified fractional-order Hindmarsh–Rose neuronal model is proposed, and the dynamics and firing activities are investigated. Some novel phenomenon can be found. First, by analyzing numerically and theoretically, the Hopf bifurcation is found to occur when the external direct current stimulus is chosen appropriately. The effects of fractional-order on the bifurcation are also studied. Second, when injecting a direct current stimulus, compared with the integer-order model, the system has more varying dynamic behaviors and firing pattern transitions. Under different external current stimulus, periodic firing patterns and chaotic firing patterns occur when fractional-order changes, but the regions of chaotic firing patterns are different. In other words, the transition mode of periodic firing and chaotic firing induced by fractional-order is different under different external current stimulus. The two-dimensional colored diagram of firing patterns is also investigated. Finally, when injecting periodic current stimulus, regular/irregular bursting, multiple spiking, regular\irregular square wave bursting, and mixed firing mods are found by setting the appropriate fractional-order, amplitude, and frequency of the external current stimulus. Some firing patterns cannot be found in integer-order models. When the amplitude is chosen at appropriate values, the region of frequency when the system displays the mixed firing modes decreases with increasing fractional-order.