Delayed feedback and detection of weak periodic signals in a stochastic Hodgkin-Huxley neuron

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Physica A

journal homepage:www.elsevier.com/locate/physa

Delayed feedback and detection of weak periodic signals in a

stochastic Hodgkin–Huxley neuron

Ergin Yilmaz

a,∗

, Mahmut Ozer

b

a_{Department of Biomedical Engineering, Engineering Faculty, Bülent Ecevit University, 67100 Zonguldak, Turkey}

b_{Department of Electrical and Electronics Engineering, Engineering Faculty, Bülent Ecevit University, 67100 Zonguldak, Turkey}

h i g h l i g h t s

• We show the impacts of autapse on the weak signal detection in a HH neuron.

• Autapse either enhances or reduces the weak signal detection in a HH neuron.

• The best weak signal detection requires a specific range of the intrinsic noise.

• The best weak signal detection requires a specific range of the autapse parameters.

a r t i c l e i n f o

Article history:

Received 10 August 2014

Received in revised form 12 October 2014 Available online 29 November 2014 Keywords:

Autapses

Self-delayed feedback Channel noise

a b s t r a c t

We study the effect of the delayed feedback loop on the weak periodic signal detection performance of a stochastic Hodgkin–Huxley neuron. We consider an electrical autapse characterized by its coupling strength and delay time. The stochastic Hodgkin–Huxley neuron exhibits subthreshold oscillations, and thus has an intrinsic time scale with the subthreshold oscillations. Therefore, we investigate the interplay of the subthreshold oscillations, coupling strength and delay time on the weak periodic signal detection. Results indicate that the delayed feedback either enhances or suppresses the weak signal detection depending on its parameters, when compared to that without the feedback. The delayed feedback augments the weak periodic signal detection for the optimal values of the intrinsic noise and the coupling strength when the delay time is close to the integer multiples of the period of the intrinsic oscillations, due to the multiple resonance among the weak signal, the intrinsic oscillations, and the delayed feedback.

We analyze the interspike interval histograms and show that the delayed feedback enhances or suppresses the weak periodic signal detection by increasing or decreasing the phase locking (synchronization) between the spiking and the weak periodic signal. We also show that an optimal phase locking is obtained when the delay time is close to the period of the intrinsic oscillations, leading a single dominant time scale in the spike trains.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

In nervous system, sensory information is converted into a tongue, so-called action potentials or spike trains by which the information is conveyed within the nervous system. Noise is omnipresent in the nervous system, and plays crucial role through the information processing. Contrary to expectations, it has therapeutic effects on the phenomena emerging in neural systems, including stochastic resonance (SR) [1–7], coherence resonance (CR) [8–12], and synchronization [13–17].

∗_{Corresponding author. Tel.: +90 372 257 4010 1273; fax: +90 372 257 2140.}

E-mail address:erginyilmaz@yahoo.com(E. Yilmaz). http://dx.doi.org/10.1016/j.physa.2014.10.096 0378-4371/©2014 Elsevier B.V. All rights reserved.

One of the most important noise sources in the neuronal system is the channel noise stemming from random open–close fluctuations of ion channels [18–20]. Effects of channel noise on the weak signal detection, coding and propagation efficiency of single neuron and biological neural networks have been extensively studied by using Hodgkin–Huxley (HH) neuron model with its stochastic extension [21–24]. The presence of an optimal intensity of channel noise, improving the weak signal coding efficiency [24] and detection performance of neuronal systems [22], has been reported in some studies.

Transmission of neural information between neurons occurs at synapses. There are two different, well-known categories of synapses; electrical synapses and chemical synapses. Furthermore, there is also a type of synapses, called autapses, a term introduced by Van der Loos and Glaser [25], are morphologically similar to normal synapses between neurons. An autapse occurs between dendrites and axon of the same neuron. There are numerous studies reporting the possible existence of autapses in different brain regions such as the striatum, substantia nigra, hippocampus and neocortex [25–28]. Tamas et al. showed anatomically that inhibitory interneurons in visual cortex form approximately 10–30 autapses [29]. Lübke et al. [30] presented that autaptic connections are existent in approximately 80% of the cortical pyramidal neurons, including neurons of the human brain. Bacci et al. [31] reported that in neocortical slices, fast-spiking (FS) but not low-threshold spiking interneurons of layer V exhibit inhibitory autaptic activity.

In addition to the above studies presenting the availability of autapses on different brain regions, there are also some studies investigating the effects of autapse on the neuronal dynamics. In this context, Bacci and Huguenard [32] experimen-tally demonstrated that inhibitory autaptic transmission plays a determinative role in enhancement of the spike-timing precision of neocortical inhibitory interneurons. They also showed that although layer V pyramidal neurons do not own inhibitory autapses, introducing of artificial inhibitory autapse via dynamic clamp increases significantly the precision of spiking [32]. Li et al. [33] analyzed the effects of electrical autapses on the spiking dynamics of the stochastic HH neuron in terms of inter-spike interval histograms. They observed an autapse-induced degradation in spontaneous spiking and also a specific frequency-locking mechanism in average inter-spike intervals [33]. Under the effect of chemical excitatory autapse, the firing rate of a HH neuron strongly depends on the duration of the activity of the autapse was declared by Hashemi et al. [34]. Masoller et al. [35] investigated the interaction among the subthreshold dynamics of a neuron, noise and time-delayed feedback which is induced by an electrical autapse by using a thermo-sensitive type of the HH neuron model, and showed that either the delayed feedback-induced spikes may be inhibited by noise or noise-induced spikes may be inhib-ited by the feedback depending on the feedback parameters and the noise. During gamma oscillations, autapse-induced increment in the synchrony of membrane potentials of basket cell, a kind of interneuron, in the network was observed experimentally by Connelly [36]. The electrical activity and firing pattern transitions induced by the autapses in a Hind-marsh–Rose (HR) neuron was studied theoretically by Wang et al. [37]. They obtained that the HR neuron exhibits spiking transitions from original firing pattern to another firing pattern or chaotic bursting firing pattern depending on the autapse parameters, i.e, autaptic coupling strength and delay time [37]. Mode-locking behaviors of a HH neuron, which is subjected to a sinusoidal stimulus, with an autapse was investigated by Wang et al. [38], where it is shown that mode-locking behav-iors can be controlled by the autapse, and the autaptic delay plays more decisive role than the autaptic coupling strength on the emergence of these behaviors.

On the other hand, autaptic transmission causes a time-delayed feedback mechanism due to the axonal conduction delay time at the level of tenths of milliseconds [35]. In many real-world systems, including biological systems time-delayed feedback is available [39,40]. The efficiency of the delayed feedback mechanism on the controlling of chaos or turbulence by stabilizing unstable periodic orbits of the chaotic attractor has been known [41]. It is reported that the time-delayed nonlinear feedback can control the coherence resonance of the Hopf oscillators by causing the transition from the spiking regime to largely bursting regime [42]. In the context of the delay, there are numerous studies which present the effect of time delay on the phenomenon emerging in complex networks [43–48]. Wang et al. [43] reported that a finely tuned delay length can remarkably augment synchronization in scale-free networks of Rulkov neurons. Impacts of electrical and chemical coupling on the synchronization transition and pattern formation of delay coupled Morris–Lecar neurons were investigated in small-world networks [44]. Complex synchronous behavior of HH neurons in interneuronal networks with delayed coupling was analyzed by Guo et al. [45]. They demonstrated that the delayed inhibitory coupling can induce a transition from regular to mixed oscillatory patterns at a critical value of the delay time [45]. The spatial coherence resonance was investigated in HH neuronal networks characterized with information transmission delay, and maximally ordered spatial dynamics were obtained for short delay lengths in an intermediate noise range [46]. In a scale-free network of different neuron models, stochastic multiresonance phenomenon emerging when the delay time equals to the integer multiple of the period of the pacemaker’s was demonstrated in Refs. [47,48].

However, despite the above studies demonstrating the effects of autapses on the temporal response of a neuron, their functional role in information processing and coding is still not clearly known. To our knowledge, there is no any study attempting to address the effects of autapse on the weak periodic signal detection performance of the HH neuron in the presence of intrinsic ion channel noise. Therefore, here our aim is to analyze the effects of autapse, specifically electrical autapse, on the weak signal detection capability of the stochastic HH neuron.

2. Model and methods

To investigate the neuronal response to weak signals in the presence of an autapse, we use classical Hodgkin–Huxley model [49] which exhibits many of the firing behaviors of a real biological neuron. In deterministic HH neuron model [49],

time evaluation of the membrane potential of a neuron is given as follows: Cm dV dt

+

g max K n 4

₍

_V

₋

_E K

) +

gNamaxm 3_h

₍

_V

₋

_E Na

) +

gL

(

V

−

El

) =

Iext

.

(1)

In Eq.(1), Cm

=

1

µ

F

/

cm2is the capacity of the cell membrane. V is the membrane potential of the neuron. EK

= −

77 mV,

ENa

=

50 mV and El

= −

54

.

4 mV are the reversal potentials for the potassium, sodium and leakage current, respectively.

gmax

K

=

36 mS cm

−2_{and g}max

Na

=

120 mS cm

−2_{respectively denote the maximal potassium and sodium conductance, when}

all ion channels are open. In the model, the leakage conductance is assumed to be constant, gL

=

0

.

3 mS cm−2. The gating

variables m, n and h in Eq.(1)represent the mean ratios of the open gates of the specific ion channels. The factors n4and m3h

are the mean portions of the open potassium and sodium ion channels within a membrane patch, and their dynamics are controlled by voltage-dependent opening and closing rates

α

x

(

V

)

and

β

x

(

V

)

(x

=

m, n, h) [49]. The dynamics of the mean

fractions of open gates with the rate functions of

α

and

β

given as follows: dx dt

=

α

x

(

V

) (

1

−

x

) − β

x

(

V

)

x

,

x

=

m

,

n

,

h (2a)

α

m

(

V

) =

0

.

1

(

V

+

10

)

1

−

exp

[−

(

V

+

40

)/

10

]

(2b)

β

m

(

V

) =

4 exp

[−

(

V

+

65

)/

18

]

(2c)

α

h

(

V

) =

0

.

07 exp

[−

(

V

+

65

)/

20

]

(2d)

β

h

(

V

) =

1 1

+

exp

[−

(

V

+

35

)/

10

]

(2e)

α

n

(

V

) =

0

.

01

(

V

+

55

)

1

−

exp

[−

(

V

+

55

)/

10

]

(2f)

β

n

(

V

) =

0

.

125 exp

[−

(

V

+

65

)/

80

]

.

(2g)

The above neuron model described by Eqs.(1),(2)do not include ion channel noise, and valid only when large number of ion channels is regarded. Hence, the model does not reflect the neuronal dynamics if the ion channel number is small. To capture the effects of channel noise in the neuronal dynamic Langevin generalization for the dynamics of the gating variables is used as below [50]:

dx

dt

=

α

x

(

V

) (

1

−

x

) − β

x

(

V

)

x

+

ζ

x

(

t

) ,

x

=

m

,

n

,

h

.

(3)

Here,

ζ

x

(

t

)

is independent zero mean Gaussian white noise sources whose autocorrelation functions are given as below [50]:

⟨

ζ

_m

(

t

)ζ

m

(

t′

)⟩ =

2

α

m

β

m NNa

(α

m

+

β

m

)

δ(

t

−

t′

),

(4a)

⟨

ζ

_n

(

t

)ζ

n

(

t′

)⟩ =

2

α

n

β

n NK

(α

n

+

β

n

)

δ(

t

−

t′

),

(4b)

⟨

ζ

_h

(

t

)ζ

h

(

t′

)⟩ =

2

α

h

β

h NNa

(α

h

+

β

h

)

δ(

t

−

t′

).

(4c)

Here, NNaand NKrepresent the total number of the ion channels for the sodium and potassium channels in a given membrane

patch, and calculated as NNa

=

ρ

NaS, NK

=

ρ

KS where

ρ

Na

=

60

µ

m−2and

ρ

K

=

18

µ

m−2are the sodium and potassium

channel densities, respectively [49–51]. S represents the total cell membrane area. It is easily seen that channel noise is inversely proportional to cell membrane area S, meaning that the larger the cell membrane area, the weaker the channel noise intensity.

In Eq.(1), Iext

=

Istim

−

Iaut, where Istim

=

A sin

(ω

t

)

is the subthreshold sinusoidal stimulus, representing the weak

signal, with the frequency

ω =

0

.

3 m s−1and amplitude A

=

1

µ

A cm−2. In this study, we consider an electrical autapse modeled with equations of Iaut

=

κ[

V

(

t

) −

V

(

t

−

τ)]

. This autaptic delayed stimulus is proportional to the difference of

the membrane potential at time t and

(

t

−

τ)

[33].

κ

denotes the autaptic coupling strength, and

τ

represents the autaptic time delay, which occurs because of the finite propagation speed during axonal transmission.

To quantitatively determine the correlation between the weak periodic signal and the output activity of the HH neuron in the presence of both channel noise and an autapse, we calculate the Fourier coefficients to measure the correlation between the weak signal frequency

ω =

2

π/

trand the temporal activity of the neuron as follows [22,52]:

Qsin

=

ω

2n

π



2nπ/ω

0

Fig. 1. Weak periodic signal detection performance of the HH neuron with and without an autapse. It is obvious that an autapse can prominently increase the detection performance of the system for proper values of the autapse parameters (ω =0.3 m s−1_{, A=1µA/cm}2_.)

Qcos

=

ω

2n

π



2nπ/ω 0 2V

(

t

)

cos

(ω

t

)

dt (5b) Q

=



Q_sin2

+

Q2 cos

.

(5c)

We calculate the Q values during n

=

300 periods (tr) of weak periodic forcing. To ensure statistical accuracy, the Q values

in all the figures below are obtained by averaging over 20 different realizations for the given parameter sets. It is noteworthy that in calculations, we consider enough transient time which is bigger than the autaptic delay time, before the calculations of the Q .

3. Results and discussions

First, we investigate the effect of the delayed feedback on the weak signal detection performance of the single HH neuron subject to a subthreshold periodic forcing with a frequency of

ω =

0

.

3 m s−1_{(the period T}

_≈

_{21 ms) and the amplitude}

A

=

1

µ

A

/

cm2_{. To do so, we numerically calculated the Fourier coefficient Q for different time delays and coupling strengths}

depending on the cell size S plotted inFig. 1. We also calculated the Fourier coefficient Q in the absence of the delayed feed-back, and also plotted inFig. 1.

It is clearly seen that in the absence of the delayed feedback (bold black line), the Q values first takes small values for small cell sizes (strong noise strengths) and increases with the increasing of S (decreasing the noise strength) and reaches a maximum, then decreases for further increase in S, indicating the clear signature of SR phenomenon. This pure resonance with respect to the channel noise (S) has already been reported in previous studies [22–24,51]. These results also demonstrate the presence of an optimal ion channel noise level (or, equivalently, the presence of an optimal cell size

S

=

16

µ

m2_{) to efficiently detect the weak input signal. However, the delayed feedback either enhances or suppresses the}

weak signal detection depending on the feedback parameters. When the delay time

τ =

20 ms, SR behavior still exists, the weak signal detection is increased by the feedback mechanism, and this increase is augmented by an increase in the coupling strength. The maximum of the Q curves occurs again around S

=

16

µ

m2_{, indicating that the best capability of}

the weak signal detection of the HH neuron requires a specific range of the intrinsic noise. However, when

τ =

15 ms, it suppresses the weak signal detection, comparing to the case without the delayed feedback. Here, the suppression becomes stronger as the coupling strength increases.

We then examined how the weak signal frequency affects the weak signal detection performance of the HH neuron with a delayed feedback, we computed the Fourier coefficient Q for the feedback parameters of

κ =

0

.

05 and

τ =

20 ms, for seven different frequencies between 0.1 and 0.7 m s−1(Fig. 2).Fig. 2shows that the weak signal detection capability of the HH neuron with the feedback is highly dependent on the weak signal frequency. The optimal performance is obtained around

ω =

0

.

3 m s−1_{and S}

₌

₁₆

_µ

_m2_{. We also found a second, but smaller peak for the same cell size but at}

_{ω =}

₀

_.

_{6 m s}−1_{, thus}

the second harmonic of

ω =

0

.

3 m s−1.

We know that an intrinsic time scale exists in neurons with subthreshold oscillations [53]. Yu et al. [54] found that the HH neurons exhibit intrinsic oscillations of period Tosc

≈

21 ms (

ω

osc

=

0

.

3 m s−1). Ozer et al. [55] showed that the collective

temporal coherence for weak periodic forcing on the small-world HH neuronal network peaks when the frequency of the weak signal matches that of the intrinsic subthreshold oscillations. They also found a smaller peak in the temporal coherence

Fig. 2. The dependence of Q on channel noise for seven different values of the weak signal frequency,ω, for the autapse parameters ofκ =0.05 and

τ =20 ms. It is clearly seen that weak signal detection capability of the HH neuron with delayed feedback is strongly dependent on the weak input signal frequency. If the frequency of the weak input signal equals to the intrinsic oscillation frequency (ω =0.3 m s−1_{) of the HH neuron, the detection} performance of the system is high.

Fig. 3. Contour plot of Q in S–τparameters domain for different autaptic coupling regimes. (Left panel) weak coupling,κ =0.05; (Middle panel) moderate coupling,κ = 0.25; (Right panel) strong coupling,κ = 0.7. It is seen that regardless of the coupling strength value autaptic delay induced multiple stochastic resonance phenomenon can be obtained. But, the weak periodic signal detection performance of the system is weaker than that of in the absence of the autapse for strong coupling regime. (ω =0.3 m s−1_{, A=1µA/cm}2_.)

for the weak signal close to the second harmonic of the oscillations, due to the resonance between the weak signal and the intrinsic oscillations. Our results inFigs. 1and2together indicate that the delayed feedback augments the weak periodic signal detection when the delay time is close to the period of the intrinsic oscillations, due to the multiple resonance among the weak signal, the intrinsic oscillations, and the delayed feedback.

Since the time delay and period of the intrinsic oscillations can be expected to interact nontrivially [35], we now turn to examine the impact of this interaction on the weak signal detection capability of the neuron. We calculated the change of

Q with cell size and the time delay in large range for three different values of the coupling strength, and showed as contour

plots inFig. 3.

For the weak coupling strength (left panel ofFig. 3), the Q values representing the system response to weak periodic forcing exhibits multiple stochastic resonance depending on the delay time. These resonances occur when the delay time is

Fig. 4. Effects of the channel noise and autaptic coupling strength (κ) on the Q response of the system for the optimal autaptic time delay,τ =20 ms. There is a red shaded region demonstrating the high detection performance of the system. The borders of this region extremely depend on the value of the autaptic coupling strength and channel noise. Under high channel noise, there is not any effect of the autapse on the weak signal detection performance of the stochastic HH neuron. Moreover, for intermediate and weak channel noise case (S≥4µm2_{), the autaptic coupling strength must be within a certain} range 0.1< κ <0.35 (ω =0.3 m s−1_{, A=1µA/cm}2_).

close to the integer multiples of the period of the intrinsic oscillations. The maximum of the Q curve drastically increases when the delayed feedback exists. Left panel ofFig. 3also shows that the best capability of the HH neuron for weak signal detection requires an optimal level of the intrinsic noise or cell size for a given optimal values of the delay time. For an intermediate coupling strength, middle panel ofFig. 3, we observed a similar behavior as for the weak coupling, but a smaller peak at the second harmonic of the intrinsic oscillations. But, the resonant behavior at

τ =

20 ms disappears due to that the combined effect of the feedback and periodic forcing is above threshold. Thus, the system response is dominated by the feedback and becomes independent from the channel noise providing that the membrane size is greater than S

>

4

µ

m2. However, maximums of the system response become very sensitive to delay time. On the other hand, for the strong coupling strength, right panel ofFig. 3, although we observed resonances when the delay time is close to the integer multiples of the period of the intrinsic oscillations, the detection capability of the neuron with the delayed feedback becomes poorer than the neuron without the feedback. Because, the feedback introduces a new time scale, which is incoherent with the weak input signal, to the neuronal dynamics. Thus, we showed that the delayed feedback results in a distinct increase in the weak signal detection capability of the neuron when both the time delay and the coupling strength have optimal values.

To gain detailed information about how the resonance is affected by the variations on both the autaptic coupling strength and channel noise, we calculate the Fourier coefficient Q in a wide range of the coupling strength (

κ =

0

.

01–0.99) and cell membrane size (S

=

1–1000

µ

m2) for an optimal delay time of

τ =

20 ms. The plot of Q values in a

τ

–

κ

parameter space is shown inFig. 4.

The system response exhibits SR behavior depending on the coupling strength. We found that the weak signal detection capability of the HH neuron strongly dependent on both cell size and coupling strength. There is no significant impact of the delayed feedback on the improvement of the weak signal detection for S

<

4

µ

m2_and

_{κ >}

₀

_.

_35.

In addition,Fig. 5reveals more insight in the dependence of the weak signal detection on both the coupling strength and delay time for a given optimal cell size of S

=

16

µ

m2_{. We again found multiple SR for a specific range of the coupling}

strength as inFig. 4when the delay time is close to the integer multiples of the period of the intrinsic oscillations. Together with the above findings, we may conclude that the best weak periodic signal detection requires a specific range of the intrinsic noise, coupling strength and delay time.

Finally, in order to provide an insight into the relationship between the spike timing of the HH neuron and the weak periodic signal, we calculated interspike interval histograms (ISIHs) over 10 000 ISIHs for the HH neuron without the delayed feedback and with a delayed feedback for

κ =

0

.

25,

τ =

20 ms and

κ =

0

.

25,

τ =

5 ms, for a given optimal cell size of

S

=

16

µ

m2_{(Fig. 6). In the absence of the delayed feedback,Fig. 6(a), the neuron exhibits an ISIH with a distinct peak near the}

period of the intrinsic oscillations, and smaller peaks at its harmonics, indicating cycle skipping for spiking. When the neuron has the delayed feedback and the delay time is close to the period of the intrinsic oscillations,Fig. 6(b), the ISIH has a single sharp peak at the period of the oscillations, indicating a single dominant time scale in the spike trains. However, when the delay time is far away for the period of the intrinsic oscillations,Fig. 6(c), the ISIH has relatively wider peaks at the harmonics. In sum, we investigated the impact of the delayed feedback on the weak periodic signal detection performance of a stochastic HH neuron, including the intrinsic noise from stochastic HH type ion channels where the membrane area or cell size S globally determines the intrinsic noise level. Results indicate that the delayed feedback either enhances or suppresses

Fig. 5. Effects of the autapse parameters on the Q response of the system for the optimal channel noise level, S=16µm2_{(ω =0.3 m s}−1_{, A=1µA/cm}2_). We obtain two resonance region occurring approximately at the integer multiples of the intrinsic oscillation period of the HH neuron for the specific range of the autaptic coupling strength.

Fig. 6. ISIHs of the stochastic HH neuron for S=16µm2_{. (a) In the absence of an autapse, (b) in the presence of an autapse withκ =0.25 andτ =20 ms,} (c) in the presence of an autapse withκ =0.25 andτ =5 ms. It is clearly seen that a stochastic HH neuron without an autapse exhibits different time scale in spiking behavior for S=16µm2_{in panel (a), but it exhibits single dominant time scale indicating an optimal phase locking between the spiking} and the weak periodic signal for proper values of the autapse parameters as shown in panel (b).

the weak signal detection depending on its parameters, when compared to that without the feedback. We showed that the best weak periodic signal detection requires a specific range of the intrinsic noise, coupling strength and delay time. The delayed feedback augments the weak periodic signal detection for the optimal values of the intrinsic noise and the coupling strength when the delay time is close to the integer multiples of the period of the intrinsic oscillations, due to the multiple resonance among the weak signal, the intrinsic oscillations, and the delayed feedback. This effect is prominent at weak and moderate coupling strengths, whereas it disappears for strong coupling strengths even for the optimal values of the intrinsic noise and delay time. For strong coupling strengths

κ >

0

.

35, the weak signal detection occurs worse than that without the delayed feedback. Thus, the delayed feedback suppresses the weak signal detection.

Sainz-Trapaga et al. [56] showed that the delayed feedback can increase the subthreshold oscillation amplitude, inducing threshold crossings, thus leading to feedback-induced spikes. Masoller et al. [57] also found that a weak delayed feedback can amplify the oscillation amplitude, thus the spiking is organized by the delayed feedback. Here, we showed that the delayed feedback enhances or suppresses the weak periodic signal detection by increasing or decreasing the phase locking (synchro-nization) between the spiking and the weak periodic signal. We also showed that an optimal phase locking is obtained when the delay time is close to the period of the intrinsic oscillations, leading a single dominant time scale in the spike trains.

In this paper, for simplicity, we only considered a neuron having one autapse which is modeled as electrical synapse. But in reality, one neuron may have more autaptic connections [29], and they may be chemical autapse. Therefore, in the context of weak signal detection, it will be interesting to consider a neuron having more autapses which are modeled as hybrid synapses [6]. On the other hand, since the neurons in the brain are embedded in the networks and autapses have deep relations with epilepsy [58], a lot of studies have been devoted to understand the impact of the time delay on the dynamics of complex neuronal networks [43–48,59,60]. Therefore, is worthwhile to investigate the impact of delayed feedback on the detection of weak periodic signals in stochastic neuronal networks.

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