The nervous system is a complex network of neurons, synapses, and other specialized cells, and through this system, neurons receive and transmit information between different parts of the body.^[1] The responses of neurons and the function of synapses have received considerable attention because of their status as building blocks of the nervous system.^{[2– 4]} Nearly a century ago, neuroscientists found a special synaptic structure, the autapse, which is a synapse between different parts of the same neuron.^{[5– 7]} Because of its odd structure, the self-synaptic connection has remained poorly understood. However, many recent studies found that autapses are much more widespread in the nervous system than previously thought. Autapses have been reported in various brain areas, such as the neocortex, cerebellum, hippocampus, striatum, and substantia nigra.^{[6, 8– 12]} Interestingly, about 80% of cortical pyramidal neurons have autaptic connections.^[8] Many studies have revealed that autapses are not merely curiosities, but play an authentic physiological role in the nervous system. Autapses can maintain persistent activity in the nervous system by mediating positive feedback.^[13]

Recent experiments also demonstrated that autapses are very important for the processing function of the brain.^[14] Bacci et al. recorded the activity of fast spiking interneurons in acute brain slices of juvenile rats and found that autaptic transmission increased the spike-timing precision.^[15] A somatic spike can evoke excitatory postsynaptic currents with various amplitudes through an autapse.^[11] However, the function of autapses and their contribution to information processing are still unclear.^{[14, 16]} Thus, understanding the effect of autaptic activities on the responses of a neuron is a fundamental step to elucidating the process of information transfer in the nervous system.^{[17– 20]}

In fact, natural autapses are self-feedback connections in the nervous system and may allow a unique type of self-control, similar to the feedback in other systems.^[14] Actually, feedback creates a circuit, or loop, that connects a system to itself and commonly occurs in many systems, e.g., gene regulatory networks, ^{[21, 22]} population dynamics, ^[23] and climate systems.^[24] Feedback is also used extensively to control the state of a nonlinear system, such as to stabilize periodic orbits or to control coherence resonance.^{[25, 26]} Feedback is also very commonly used in the design of electric circuit elements.^[27] In these systems, information about the past or the present influences the same phenomena in the present or future, respectively. Moreover, feedback has a marked effect on the dynamics of nonlinear systems. The frequency at the onset of the oscillation of a nonlinear system can be modified by the feedback loop.^[28] Furthermore, self-delayed feedback with a fixed delay time can suppress chaos and also control steady states.^{[29, 30]} In a simulation study, Popovych et al. showed that time-delayed feedback has the ability to desynchronize groups of model neurons.^[31]

A neuron with an autaptic connection can provide a distinctive physiological self-feedback model with many dynamic properties. Rusin et al. performed a time-delayed feedback stimulation of a group of cultured neurons and experimentally demonstrated that time-delayed feedback could cause synchronization of the action potentials of a group of neurons.^[32] Prager et al. found that time-delayed feedback can facilitate the noise-induced oscillation of a neuronal system.^[33] Autaptic-delayed feedback was found to modify the bursting in the distribution of interspike intervals of a stochastic Hodgkin– Huxley (HH) neuron.^[34] Such autaptic-delayed feedback can also reduce the spontaneous spiking activity of a stochastic neuron at the characteristic frequencies. The effect of an autapse on the spike rate of a single neuronal system shows dependence on the duration of the autapse activity.^[35] Modulation of autaptic delay feedback can cause the dynamic behaviors of a Hindmarsh– Rose (HR) neuron to switch between quiescent, periodic, and chaotic firing patterns.^[36]

In this article, we review the current studies of knowledge of the effect of an autapse on single neurons.

Scientists have long investigated the functions and properties of neurons and synapses. Neurons vary widely with different sizes and forms, and neuron responses are also involved in many dynamic behaviors. A synapse is a structure in the nervous system that permits a neuron (or nerve cell) to transmit an electrical or chemical signal to another cell (neural or otherwise).^[37] The self-synapse, or the autapse, was first described as a synapse between the axon of a pyramidal cell and its own dendrites by Van der Loos and Glaser in 1972.^[5] Before the report by van der Loos et al., other terms, such as ‘ self-excitation’ and ‘ self-sensing’ , were used to describe the self-synaptic structure. For a long time, autapses in the nervous system seemed like anatomical curiosities with questionable functional significance. However, recent experiments began to reveal how autapses might play an important role in brain function. Some reports also suggested that autapses are connected to some neural diseases.^[38] Neurobiology experiments have always focused on excitatory (glutamate-releasing) and inhibitory (GABA-releasing) autapses. To date, autaptic structures have been found in almost all parts of the nervous system.^[6] Figure 1 shows photos of the many autapses of a single rat hippocampal neuron.

Fig. 1.

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	Fig. 1. A single rat hippocampal neuron (arrowed, left) grown in culture on a ‘ microdot’ of glia (flat gray cells), showing abundant autapses labeled with an antibody (small dark spots, right).[14]

In nature, autaptic connections enable self-feedback of a neuron. Thus, in almost all current autapse studies, the model of a neuron with an autaptic connection contains a single compartment that exhibits a self-delayed feedback mechanism, as shown in Fig. 2. The delay time represents the elapsed time associated with the axonal propagation prior to the reintroduction of the signal into the neuron. The delay time is also believed to be one of the important properties of autaptic connections. Mathematical models of an autapse in previous studies of the effect of self-synapses on neurons fall into two main categories: one has linear self-coupling and describes a gap junction, and the other has nonlinear self-coupling and describes a chemical synapse, which may be an excitatory or inhibitory synapse. The specific mathematical form of the autapse depends on the chosen neuronal model.

Fig. 2.

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	Fig. 2. Schematic drawing of a neuron with autaptic feedback. The structure in the circle represents an autaptic connection between dendritic and axonal terminals.

The autaptic connection of a simplified neuron model (such as the HR neuron, FitzHugh– Nagumo neuron, Izhikevich neuron and others) can often be described by one of the following formulas:^{[35, 36, 39]}

For a conductance-based neuron model (such as the HH neuron, Morris– Lecar neuron, Connor– Stevens neuron and others), the mathematic model of the autapse is often given as follows^{[34, 35, 42, 43]}

In the following sections, we will review the two main forms of an autapse. Studies have shown that autapses that exhibit delayed feedback provide a control option for adjusting neuron responses.

Experimental observations have indicated that action potentials can occur with different firing patterns, and the primary firing patterns are spiking and bursting.^{[45– 50]} Bursting is an extremely diverse general phenomenon in firing patterns exhibited by neurons in the central nervous system and spinal cord.^{[51, 52]} Previous studies on a bursting neuron with an autapse showed the novel dynamics and the transition of firing patterns induced by an autapse.^[36]

The simplified model of Hindmarsh and Rose has turned out to capture the features of experimentally measured electrical data quite accurately, particularly for studies of the spiking-bursting behavior of neuron membrane potentials.

Without an autapse, HR neurons exhibit many dynamic behaviors, including quiescent, regular spiking, periodic, and chaotic bursting firing patterns (see Fig. 1 in Refs. [36] and [53]).

The presence of an autapse completely changes the firing patterns of the original HR neuron. The firing pattern can be adjusted from a periodic or chaotic pattern to another periodic pattern or to a chaotic bursting pattern as the autaptic parameters change, independently of the original firing pattern. Figure 3 shows the time courses of the membrane potentials of an HR neuron with an electrical autapse as an example. Importantly, the maximum action potential is increased by the electrical autapse.

Fig. 3.

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	Fig. 3. Time courses of the membrane potentials in response to different Iext with gaut = 0.5.[36] The blue curves represent the time course of the HR neuron with an autapse, and the red curves indicate that without autapse.

The ISI plot bifurcation diagrams of the HR neuron clearly show the transition between periodic and aperiodic firing. For an electrical autapse, the periodic state transits to the chaotic state exhibiting an alternating behavior as the time delay increased. With higher autaptic intensity, this alternating behavior is more noticeable and occurs more frequently. For shorter delay times, the firing pattern of the HR neuron showed periodic spiking independent of the external DC input. The neuron with an excitatory chemical autapse exhibits chaotic firing patterns in a larger area of g_aut– τ space than that of the neuron with an electrical autapse. For strong external stimuli, the chaotic region of the combinational parameters in the g_aut– τ space is enlarged. The excitatory autapse plays a positive role in generating and enhancing chaos as a whole. As neurons with an inhibitory autaptic connection, the chaotic spiking of the HR neuron can be decreased and suppressed. With the proper inhibitory autaptic parameters, the HR neuron could be driven to a resting state. Figure 4 shows the bifurcation of the ISIs of an HR neuron for the three types of autapses.

Fig. 4.

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	Fig. 4. Bifurcation diagram of the interspike interval of an HR neuron with an autapse versus the autaptic delay time.[36] From left to right, the panels show the results of the electrical autapse, excitatory chemical autapse, and inhibitory chemical autapse.

Without an autapse, the firing pattern of an HR neuron switches from silent to periodic bursting and then to chaotic firing with period-doubling bifurcation as the DC current increased.^{[54– 57]} For neurons with an autapse, the firing pattern can switch into any other firing pattern, regardless of the original pattern. As a whole, there are two main approaches to transition from a periodic to a chaotic firing pattern: discontinuous and continuous. The interspike interval return map is a useful approach to characterizing the transition to chaos. Figure 5 shows an example of the two approaches for the transition of a neuron to chaos. In the discontinuous transition, the neuron has a periodic bursting firing pattern because the delay time is short. When the delay time increases, the system suddenly enters a chaotic firing state. In the continuous transition, the neuron first displays a periodic firing pattern. As the autaptic parameters change, the number of spikes in the single burst as well as the period of the periodic bursting increase almost continuously, and the neuron transitions into the chaotic state.

Fig. 5.

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	Fig. 5. An example of the two ways of transition (the discontinuous transition [upper] and the continuous transition [bottom]) into chaos from periodic firing.[36] External stimuli is set as Iext = 2.67.

When excited by a periodic stimulus, neurons respond with various mode-locking firing patterns and quasi-periodic states.^{[42, 58– 61]} When an autapse is present, the mode-locking firing of a neuron can be switched to another state, depending on the chosen autapse parameters. An autapse provides self-feedback and contributes an additional timescale to the dynamic neuronal system. Thus, the neuron– autapse system may exhibit very complex dynamics, due to the interplay between autaptic delayed feedback, an external periodic stimulus, and the intrinsic activity of the neuron.

In previous studies, the model neuron– autapse system often contained an HH neuron and an autapse. The HH model is a conductance-based model that describes how action potentials in neurons are initiated and propagated.^[62] In this section, we review the mode-locking firing pattern of an HH neuron with an electrical autapse that has the same mathematical form as Eq. (1).

Without an autapse, the mode-locking behaviors depend on the values of the stimulus frequencies and amplitudes. As shown in Fig. 1 in Ref. [42], the Arnold tongues in the frequency and amplitude space show the overall features of various phase-locked states and provide the different p : q (denoting output action potentials per input spikes) mode-locking regions. Reference [58] also shows the bifurcation mechanisms that create the boundaries of the complex mode-locking structure.

The presence of an autapse substantially modifies the mode-locking patterns of the neuron. For the same sinusoidal stimulus, the neuron with an autapse can fire more or fewer action potentials than that of a neuron without an autapse. The activities of the neuron can also be driven into sub-threshold oscillation. The electrical autapse displays a regulating function that controls the mode-locking firing of the neuron. The time courses of the membrane potentials of the neuron with and without an autapse are shown in Fig. 2 of Ref. [42]. With an autapse, the modification of the mode-locking firing pattern depends on the autaptic conductivity and the delay time. When the synaptic conductivity is small, the mode-locking p : q is similar to that without an autaptic connection. When the synaptic conductivity is large, the neuron displays very complex mode-locking firing. The p : q value of mode-locked firing increases from zero to a very large value when the frequency of the sinusoidal current increases from zero to the “ threshold” frequency (the minimal input frequency for which the neuron fires an action potential given a fixed input amplitude). For further increases in the frequency of the sinusoidal current, the p : q value of mode-locking firing decreases. In the case of extremely strong autaptic conductivity [see Fig. 3(d) in Ref. [42]], the autaptic activities completely disrupt the original mode-locking firing of the neuron without an autapse.

For continuous changes in the delay time, the model-locked firing of the neuron is also quite interesting. Figure 6 shows the dependence of the mode-locking state on the delay time for different input frequencies and autaptic conductivities. For weak autaptic conductivities [Fig. 6(a)], the p : q values of mode-locking are similar to those without self-feedback for the short delay time. As the delay time increases, the values of p : q mode-locking begin to fluctuate. The smaller the input frequency is, the larger the resulting fluctuation. For strong autaptic intensities [shown in Figs. 6(b)– 6(d)], the p : q values of mode-locking decrease with increasing delay time in a stepwise manner. With further increases in the delay time, the p : q values of mode-locking suddenly jump to a very large value (larger than that without autaptic self-feedback) and then decrease smoothly. As the delay time increases, the p : q value of mode-locking of a neuron with a high autaptic intensity exhibits periodic behavior.

Fig. 6.

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	Fig. 6. Mode-locking of responses of an HH autaptic neuron with different delay times.[42] The lines with different markers denote the firing rate of the neuron excited by different input frequencies.

With autaptic self-feedback, the responses of an HH neuron to a sinusoidal stimulus can have higher or lower p : q mode-locking responses than that of a neuron without an autapse. These dynamical behaviors depend on the autaptic intensity and the delay time. Moreover, the presence of an autapse increases the range of values for which the HH neuron spiking is locked with the sinusoidal current. When the autaptic intensity is weak, the mode-locking behaviors show no marked changes. That is, an autapse with a weak intensity will have a negligible effect on the response of the neuron. For stronger autaptic intensities, the p : q value of the mode-locking increases and then decreases as the delay time increases, exhibiting nearly periodic behaviors. Thus, for sufficiently strong autaptic intensity, changing the delay time provides better regulation of mode-locking than does changing the autaptic intensity. The autaptic connection may also provide a control option for adjusting mode-locked firing in a neural information process.

Neuronal noise is random electrical fluctuations within neuronal networks and affects the patterns of neural activity in a determinant way.^{[75– 78]} In the presence of noise, autapses can also substantially affect the response of a neuron. In this section, we review previous studies on the interplay of autapses and noise on the firing of an autaptic neuron.

Considering the subthreshold dynamics of a neuron with interaction between autaptic-delayed feedback and noise, Masoller et al. investigated the firing patterns of an HH model of a thermoreceptor neuron with an electrical autaptic feedback in the presence of Gaussian white noise.^[79] In their studies, the neuron displays only subthreshold oscillations in the absence of feedback and noise. Their results show that the interaction among weak autaptic feedback, noise, and the subthreshold intrinsic activity is nontrivial. The subthreshold oscillation amplitude can be enhanced by the autapse in the presence of external noise, and this enhancement is more pronounced for certain delay values. For negative autaptic delay feedback, the firing rate can be lower than that of the noise-free situation, depending on the delay. This is because noise inhibits feedback-induced spikes by driving the neuronal oscillations away from the firing threshold. For positive autaptic delay feedback, there are regions of delay values where the noise-induced spikes are inhibited by the feedback; in this case, the autaptic feedback drives the neuronal oscillations away from the threshold.

Li and his co-workers analyzed the effects of electrical autapses on the spiking dynamics of a stochastic HH neuron, ^[43] considering the stochastic gating of ion channels or the so-called intrinsic channel noise is considered. They found that the delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution, exhibiting a delay-induced reduction in the spontaneous spiking activity at characteristic frequencies. For small numbers of ion channels, the channel noise is sizable and the excitatory dynamics remain practically unaffected by the delay. However, smaller noise levels and stronger autaptic intensities induce different synchronization phenomena between the delay time and the intrinsic time scales. The delay time and the intrinsic timescales determine the number of spikes that will be induced and become subsequently locked during one delay epoch.

Recently, Yilmaz and Ozer showed that the electrical autaptic delayed feedback either enhances or suppresses the weak signal detection, depending on the parameters of autapse and channel noise.^[80] When the delay time is close to integer multiples of the period of the intrinsic oscillations, the autapse enhances the weak periodic signal detection for the optimal values of the intrinsic noise and the autaptic intensity. The system response also exhibits stochastic resonance behavior, depending on the autaptic intensity. Moreover, the weak signal detection capability of the HH neuron is strongly dependent on the cell size and autaptic strength.

Self-delayed feedback significantly affects nonlinear dynamic systems because such self-feedback loops introduce a new timescale into the dynamics of the system.^{[81, 82]} The system of a neuron with an autapse contains two timescales: the intrinsic timescale of the neuron and the timescale of the autapse. However, the situation is more complex in the nervous system, because there are many more timescales, including the timescales of the neurons, the synapse, the environment, and the autaptic connections. This complexity raises the question of how the activity time of an autapse influences the activities of coupled neurons in the nervous system.

Autapses provide self-feedback circuits that are common in the nervous system. Since the naming of autapses, many experimental studies have revealed that autapses play important roles in brain function.^{[6, 11, 14, 15, 17, 18, 20]} The current theoretical studies on the topic of autaptic connections also reported the importance of autapses and the many novel dynamic behaviors induced by the autapse.^{[32, 34– 36, 42, 43]} Autapses provide a control option that can sufficiently adjust the firing behaviors of a neuron for any form of input stimulus, regardless of the neuron type.

Autapses offer a new mechanism for switching between quiescent, periodic and chaotic firing patterns in bursting neurons.^[36] Additionally, the nervous system responds rapidly to an external stimulus based on the transition between neuron firing patterns.^{[40, 83]} Thus, the results of the firing pattern transition induced by an autapse also indicate that an autapse could act as an efficient tool for controlling the transition among different relevant neuronal activities in the nervous system.

For an HH neuron with an autapse, the firing frequency and interspike interval distributions of the output spike train show periodic behavior when the delay time is increased.^[43] When specific autaptic parameters are chosen, the response spike trains are nearly regular, and the ISI distribution covers a small area, even with a highly random input. This phenomenon emphasises the autapse-induced filtering behaviors of the neuron. These results about the autapse are useful for studying the control of a nonlinear system.

In neuronal systems, an autapse can take the form of recurrent excitation, i.e., the discharge of a neuron, possibly after passing through the axon, can subsequently induce an excitatory response in the same neuron. This self-excitation mechanism is important for maintaining persistent activity, particularly the feeding behavior in sea slugs. Leonel et al. also reported that such processes play functional roles in amplifying activity in neuronal assemblies, causing reverberating activity, inducing some form of memory, or generating rhythmic patterns, such as those in central pattern generators.^[84] The previous experimental data also indicate that the brain tissue expresses a novel form of self-inhibition, namely autaptic inhibitory transmission in Fast-Spike (FS) cells slow self activities in interneurons.^[85] Ma et al. also investigated the activities of a two-dimensional neural network containing autapses with different time delays and found that the autapses induced many novel collective behaviors.^[86] The effect of autapse on the synchronization of a neural network has also been conducted in a group of HH neurons with small world network structure.^[87] It was found that the neurons exhibit synchronization transitions as autaptic delay feedback is varied, and fine synchronized network activities occur when an optimal autaptic strength is chosen. Alberto Bacci and his colleagues studied the autaptic self-inhibition of basket cells and the role of the autaptic connections of disinhibition within cortical circuits and argued that autaptic feedback could have a dual function in temporally coordinating parvalbumin basket cells during cortical network activity.^[88] Another study also indicated that fast spiking neurons with autapses showed the strongest asynchronous release in brain slices obtained from patients with intractable epilepsy, and that such discharges may be involved in generating and regulating network activities, including epileptic activity.^[38]

Although there are many studies on the topic of the autapse, the precise function of autapses and their contribution to information processing remain unclear. There are also many open questions both in the experimental and theoretical areas: 1) What is the role of autapses in co-ordinating network activities? i.e., FS cell autapses in adjusting fast network synchrony. 2) What are the molecular mechanisms underlying autaptic asynchronous release and what is its relevance during physiological and pathological network activities? 3) What are the functions of autapses in the information propagation in the neural circuit and neural network? Addressing these questions will also facilitate our understanding of the fundamental mechanisms governing several core functions of cortical activities.