- Email: [email protected]
A positive role of noises in accurate detection of time difference by electrosensory system of weakly electric fish
Neurocomputing 32}33 (2000) 855}862
A positive role of noises in accurate detection of time di!erence by electrosensory system of weakly electric "sh Yoshiki Kashimori*, O. Hoshino, Takeshi Kambara Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo, 182-8585 Japan Accepted 13 January 2000
Abstract Weakly electric "sh Eigenmannia can discriminate with an accuracy of the order of 1 ls whether the phase of its own EOD signal is advanced or delayed compared with that of its neighbor's signal. To clarify the neural mechanism accomplishing the hyperaccurate discrimination, we present a neural model of the torus. The time di!erence between a!erent nerve spikes arising from di!erent sites on the body surface is detected by the two-dimensional array of small cells in the torus. The phase advance and delay are detected by integrating the outputs of small cells. The hyperaccurate detection is accomplished by the stochastic resonance induced by the spatio-temporal integration of the outputs including noises. 2000 Elsevier Science B.V. All rights reserved. Keywords: Weakly electric "sh; Jamming avoidance response; Neural modeling; Noise; Phase modulation
1. Introduction Many neural systems encode information in the time domain with microsecond accuracy [4,9,10]. This time domain would seem too short for single neurons to encode or resolve stimuli, because neural events occur in a millisecond, rather than a microsecond, time scale. It is not yet clear how the temporal information is processed within such a short time domain. In order to clarify the neural mechanism for the detection of time di!erence with accuracy of microsecond, we study the neural
* Corresponding author. Tel.: #81-424-43-5470; fax: #81-424-43-5470. E-mail address: [email protected] (Y. Kashimori). 0925-2312/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 2 5 3 - 8
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system for jamming avoidance response (JAR) in weakly electric "sh, because the temporal information processing of JAR has been actively studied based on physiological and ethological experiments [2}4]. The weakly electric "sh Eigenmannia analyzes the distortion of electric "eld around its body surface to perceive its environment [1]. When two "shes, whose electric organ discharges (EOD) show similar frequencies, come close, each "sh shifts its own EOD frequency so as to increase the frequency di!erence between them [4]. This behavior is called jamming avoidance response (JAR). Eigenmannia uses two cues for JAR, modulations of EOD amplitude and of EOD phase which are induced by mixing of its own EOD with its neighbor's EOD. In the initial stage, the two kinds of modulations of jamming signal are encoded by two types of electroreceptors, P- and T-type receptors. The signals encoded are conveyed in parallel to the higher-order nucleus, electrosensory lateral-line lobe (ELL) of the hindbrain and torus semicircularis (TS) of the midbrain. Neurons in ELL detect the modulation of stimulus amplitude, while neurons in TS detect the modulation of stimulus phase. The a!erent signals into which both the modulations are encoded are integrated over the electroreceptors distributed on the body surface and converged into deeper laminae of the torus. Using the signals converged, the "sh can discriminate whether its own EOD frequency is larger or smaller than its neighbor's EOD frequency. Behavioral experiments have demonstrated that these "shes detect the phase di!erence between their own EOD and jamming EOD down to 1 ls [4,7,8]. The phase is represented by the timing of zero-crossing of EOD which are represented by impulses of T-type receptor cells. Each small cell in the lamina VI of the TS is sensitive to coincidence of arrival times between the two impulses arising from the T-receptor cells locating at di!erent sites on the "sh body surface. The two-dimensional array of small cells detects the time di!erence with an accuracy of 10 ls [2,3]. The accuracy corresponds to of the usual neural relaxation time (&1 ms). It is remarkable that the system constructed by neurons with much longer processing time can achieve such a short temporal resolution. The neural mechanism realizing the hyperacuity has not yet been made clear. In the present study, we present a neural network model of the torus for the detection of phase advances and delays, which has been made based on the physiological and anatomical data. The network model discriminates phase advance and delay by detecting the time di!erence between a!erent nerve spikes arising from di!erent sites on the body surfaces and integrating the spatio-temporal "ring pattern of the small cells in the lamina during JAR. The single-chain model including a single giant cell axon detects the phase advance and delay up to 60 ls. The detection ability is much increased by integrating the outputs of small cells in the two-dimensional arrays consisting of multiple chains shown in Fig. 1. The discrimination accuracy of the lamina model is noticeably improved by adding noises with optimal intensity to the a!erent signals with a smaller time di!erence. The underlying mechanism is similar to that of stochastic resonance, but the method picking up only meaningful signals from noise-enhanced response spikes is new.
Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862
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Fig. 1. A neural network model of the torus for detection of phase advance and delay.
2. A network model of the torus for detection of phase advances and delays (di4erential phase coder, DPC) Neurons in the torus semicircularis, or small cells, compare timings of a!erent nerve spikes arising from two di!erent parts of the body surface. At early stage of JAR, the phase of EOD wave is encoded as the timing of zero-crossings of EOD wave by T-receptors, and the timings are represented by a!erent nerve spikes. The a!erent signals are converged to giant cell bodies and small cell dendrite in the torus, via spherical cell in the medulla, as shown in Fig. 1 [4]. Each small cell receives two kinds of signals from its own dendrite and from a single giant axon, respectively, which are received signals from di!erent body surface areas as shown in Fig. 1. A small cell detects the coincidence of arrival between the two signals. We proposed a cellular model of a small cell which acts as a coincidence detection unit [6]. We showed that the small cell model detects the coincidence between two input spikes with an accuracy of tens of ls. However, to perform a correct JAR, it is required [4] to detect whether the signal from area A is advanced or delayed compared with that from area B. To detect the phase advances and delays, we presented here a network model which consists of multiple linear arrays of the small cells (coincidence detection (CD) chain) in lamina 6 and two types of neurons, A- and D-neurons in lamina 8b, as shown in Fig. 1. We call hereafter the network model di!erential phase coder (DPC). A CD chain consists of small cells and delay lines (bold line from a giant cell shown in Fig. 1). Each small cell of CD chain receives a spike with time delay depending on the distance between the small cell and the giant cell. The small cell receives simultaneously the spike from the spherical cell, which has a time delay common to all the small cells. Because it has been suggested [2,3] that the torus does not have the ordered map structure with respect to time delay such as the nucleus laminaris of barn owl, the time
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of spike propagation between neighboring small cells was set in the model to be di!erent depending on the position of the small cell along the CD chain. The outputs of small cells in each CD chain are integrated by A- and D-neurons of a deeper laminar 8b, as shown in Fig. 1. The mathematical description of a small cell model is given in Ref. [6]. The membrane potentials, < , and the outputs, ; , of A- and D-neurons 7 7 (>"A, D) are determined by 1 ,* , d< 7 "!< # q w z (>"A, D), (1) 7 N 7 GI GI 7 dt * I G 1 z " , (2) GI 1#exp(!(v !h )/e) GI I w "w exp(!(i!1)/p )#a , (3) " GI I " w "w exp(!(N!i)/p )#a , (4) GI I 1 ; " , (5) 7 1#exp(!(< !h )/e ) 7 7 7 where w is the strength of connection from the ith small cell of the kth CD chain to 7 GI >-neuron and decays exponentially depending on the distance between the small cell and >-neuron (>"A, D). z and v are the output and the membrame potential of GI GI the ith neuron in the kth CD chain, respectively, and ; is the "ring rate of neuron >. 7 In the present paper, small cells in each CD chain were arranged linearly with an equal distance for simplicity. The "ring rates of A- and D-neurons increase with increasing the magnitude of phase advance and delay because of the exponential decay of the synaptic strength w . 7 GI 3. Results 3.1. Ability of DPC network to detect phase advances and delays We investigated the detection ability of the DPC model for phase advance and delay in the case where the model has a single CD chain. We used here two types of jamming signals received at two di!erent sites on body surface; one signal whose distortion from its own EOD due to jamming is very small and the other signal whose distortion is large. The former signal is transmitted to the small cell soma, while the latter is transmitted to the small cell dendrite. The spike trains of peripheral nerve arising from these signals were calculated by T-receptor model [5]. The time disparity between two spikes from a giant cell and a spherical cell is changed sinusoidally as shown in Fig. 2A(ii), where the maximum time disparity (¹D ) is 160 ls. Then, the sites of small cell excited the most strongly by the two
signals are changed as the magnitude of disparity is changed, as shown in Fig. 2A(iii).
Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862
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Fig. 2. (A) (i) Jamming signal. (ii) Temporal variation of time disparity detected by T-a!erent model. The time di!erence was measured with reference to the timing of spike from a giant cell. (iii) The spatio-temporal variation of "ring rates of CD units linearly arranged along the vertical axis. The firing rates are plotted in the direction perpendicular to the sheet. This is a two-dimensional projection of the rates. (iv) Temporal variations of "ring rates of A- and D-neurons. (B) The dependence of the di!erence *o"o !o between " o and o on the maximum shift N of the input spikes due to noise. (C) The dependence of the "
ampli"cation ratio r of the di!erence of average "ring rates on ¹D . The `subthresholda and `supra6
thresholda indicate the region of ¹D which is subthreshold and suprathreshold for a clear detection in
the absence of noise, respectively.
The sites of the small cells with the maximum "ring rate move from the delay side to the advance side in the linear array of small cells. The small cells only in a de"nite region of CD chain can "re actively as seen in Fig. 2A(iii), and the number of "ring cells corresponds to the accuracy for detection of time disparity by the CD chain. The accuracy accomplished by the single linear chain of the small cells is about 100 ls, because the time delay between a small cell and its neighbor is set to be 10 ls. Fig. 2A(iv) shows the "ring rates of A- and D-neurons. In the region of phase advance (negative time disparity), the "ring rate of A-neuron is larger than that of D-neuron. On the other hand, in the region of phase delay (positive time disparity), the "ring rate of D-neuron is larger than that of A-neuron. Thus, A- and D-neuron discriminates correctly the phase advance and delay, respectively.
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To evaluate the detection ability of the DPC chain model, we calculated the average "ring rates of A- and D-neurons for the spike patterns with various values of ¹D .
The average "ring rates of A- and D-neurons, o6 and o6 (X"dl, ad) were calculated " in the periods of phase delay (0}100 ms) and phase advance (100}200 ms) of one cycle of the jamming beat pattern. The di!erence between the average "ring rates of A- and D-neurons in the delay region is decreased as ¹D is decreased. For ¹D less than
60 ls, the average "ring rate, o , noticeably decreases. The chain model can discrimi" nate correctly the phase advance and delay for ¹D larger than 60 ls.
3.2. A positive role of input noises in hyperaccurate detection To investigate the role of input noise in the improvement of the detection acuity for the DPC network model with multiple CD chains shown in Fig. 1, we calculated the di!erence in the average "ring rate between A- and D-neurons under the application of input noises with various intensities. The result is shown in Fig. 2B. The interspike intervals into which phase modulation of a jamming signal is encoded are shifted randomly by values within the region from !N to N . As the width of random
shift increases, the di!erence in the average "ring rate between A- and D-neurons increases and then slightly decreases. It becomes maximum at a certain value of the width (N "60 ls). The detection ability of phase advance and delay becomes
higher with increasing of the di!erence of the "ring rate. To evaluate the e!ect of noise on the improvement of detection ability, we calculated the ratio, r "*o6 /*o6, where *o6 (X"ad, dl) is the di!erence in the 6 , , average "ring rate between A- and D-neurons under the application of the optimal noise whose width is N "60 ls, and *o6 is that in the absence of noise. r means
6 the ampli"cation ratio of the detection ability under the application of noise. The result is shown in Fig. 2C. The ratio increases as ¹D decreases. For ¹D smaller
than 60 ls, the ratio noticeably increases, while for ¹D larger than 60 ls, the ratio
remains constant. It appears that the optimal noise enhances more e!ectively the sensitivity of A- and D-neurons for smaller ¹D , which are subthreshold for the
clear detection in the absence of noise. The mechanism of improvement may be similar to that in the stochastic resonace [11]. When the input spike train with the small ¹D (440 ls) does not include the
noise, the "ring region of the CD chain slightly #uctuates temporally within the region of CD layer. The shift is within the subthreshold. When the input spike trains include the noise, the "ring region of the CD chain is shifted more largely to the delay or advance side of the CD chain. The shift is stochastically out of the threshold. Because the magnitude of spike interval shift is chosen randomly from the range from !N to N , the "ring region is shifted more to the delay region in one CD
chain, but more to the advance region in another CD chain. When the noises with optimal width are added to the input spike train with a subthreshold time di!erence, the noises are able to enhance the small shift of "ring zone relevant to the phase advance and delay, because the optimal noise enhances cooperatively the subthreshold shifts over the CD chains.
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References [1] J. Bastian, Electrolocation, Behavior, Anatomy, and Physiology, in: T.H. Bullock, W. Heiligenberg (Eds.), Electroreception, Wiley, New York, 1986, pp. 577}612. [2] C.E. Carr, Processing of temporal information in the brain, Ann. Rev. Neurosci. 16 (1993) 223}243. [3] C.E. Carr, M.A. Friedman, Evolution of time coding systems, Neural Comput. 11 (1999) 1}20. [4] W. Heiligenberg, Neural Nets in Electric Fish, Chaps. 3 and 4, MIT Press, Cambridge, 1991. [5] Y. Kashimori, M. Goto, T. Kambara, A Model of P- and T-electroreceptors of weakly electric "sh, Biophys. J. 70 (1996) 2513}2526. [6] Y. Kashimori, T. Kambara, A neural mechanism for hyperacuity in jamming avoidance response of electric "sh Eigenmannia, Proceedings of 1997 IEEE International Conference on Neural Networks Vol. 1, 1997, pp. 332}337. [7] M. Kawasaki, G. Rose, W. Heiligenberg, Temporal hyperacuity in single neuron of electric "sh, Nature (London) 336 (1989) 173}176. [8] M. Kawasaki, Sensory hyperacuity in the jamming avoidance response of weakly electric "sh, Curr. Opin. Neurobiol. 7 (1997) 473}479. [9] M. Konishi, Listening with two ears, Sci. Am. 268 (1993) 66}73. [10] J.A. Simmons, The resolution of target range by echolocating bat, J. Acoust. Soc. Am. 54 (1973) 157}173. [11] K. Wiesenfeld, F. Moss, Stochastic resonance and the bene"ts of noise: from ice ages to Cray"sh and Squids, Nature 373 (1995) 33}36.
Yoshiki Kashimori received his Ph.D. from Osaka City University in 1985. He is a research associate in the Department of Applied Physics and Chemistry, at the University of Electro-Communications. His research interest is to clarify the neural mechanism of information processing in the electrosensory, auditory, and gustatory systems, based on the modeling of neurons and their network. He also investigates the emergence of dynamical orders in various biological systems, based on the nonlinear dynamics.
Osamu Hoshino received his Ph.D. in Biophysics from the University of Electro-Communications in 1998. He is presently an assistant professor in the Department of Human Welfare Engineering at Oita University. His research interests include the neural mechanisms of olfactory, auditory, and memory systems. He is also interested in the neural basis of cortical mapping and its role in human behavior.
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Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862 Takeshi Kambara received his Ph.D. from Tokyo Institute of Technology in 1970. He is a Professor of Biophysics in the Department of Applied Physics and Chemistry and a Professor of biological information science in the Graduate School of Information Systems, at the University of Electro-Communications. His scienti"c interests cover the neural mechanism of information processing in olfactory, auditory, visual, gustatory, and electro-sensory systems, the brain model of consciouness, and emergence of dynamical orders in various biological complex systems. His research work uses the `in silicoa method.
A positive role of noises in accurate detection of time di!erence by electrosensory system of weakly electric "sh Yoshiki Kashimori*, O. Hoshino, Takeshi Kambara Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo, 182-8585 Japan Accepted 13 January 2000
Abstract Weakly electric "sh Eigenmannia can discriminate with an accuracy of the order of 1 ls whether the phase of its own EOD signal is advanced or delayed compared with that of its neighbor's signal. To clarify the neural mechanism accomplishing the hyperaccurate discrimination, we present a neural model of the torus. The time di!erence between a!erent nerve spikes arising from di!erent sites on the body surface is detected by the two-dimensional array of small cells in the torus. The phase advance and delay are detected by integrating the outputs of small cells. The hyperaccurate detection is accomplished by the stochastic resonance induced by the spatio-temporal integration of the outputs including noises. 2000 Elsevier Science B.V. All rights reserved. Keywords: Weakly electric "sh; Jamming avoidance response; Neural modeling; Noise; Phase modulation
1. Introduction Many neural systems encode information in the time domain with microsecond accuracy [4,9,10]. This time domain would seem too short for single neurons to encode or resolve stimuli, because neural events occur in a millisecond, rather than a microsecond, time scale. It is not yet clear how the temporal information is processed within such a short time domain. In order to clarify the neural mechanism for the detection of time di!erence with accuracy of microsecond, we study the neural
* Corresponding author. Tel.: #81-424-43-5470; fax: #81-424-43-5470. E-mail address: [email protected] (Y. Kashimori). 0925-2312/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 2 5 3 - 8
856
Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862
system for jamming avoidance response (JAR) in weakly electric "sh, because the temporal information processing of JAR has been actively studied based on physiological and ethological experiments [2}4]. The weakly electric "sh Eigenmannia analyzes the distortion of electric "eld around its body surface to perceive its environment [1]. When two "shes, whose electric organ discharges (EOD) show similar frequencies, come close, each "sh shifts its own EOD frequency so as to increase the frequency di!erence between them [4]. This behavior is called jamming avoidance response (JAR). Eigenmannia uses two cues for JAR, modulations of EOD amplitude and of EOD phase which are induced by mixing of its own EOD with its neighbor's EOD. In the initial stage, the two kinds of modulations of jamming signal are encoded by two types of electroreceptors, P- and T-type receptors. The signals encoded are conveyed in parallel to the higher-order nucleus, electrosensory lateral-line lobe (ELL) of the hindbrain and torus semicircularis (TS) of the midbrain. Neurons in ELL detect the modulation of stimulus amplitude, while neurons in TS detect the modulation of stimulus phase. The a!erent signals into which both the modulations are encoded are integrated over the electroreceptors distributed on the body surface and converged into deeper laminae of the torus. Using the signals converged, the "sh can discriminate whether its own EOD frequency is larger or smaller than its neighbor's EOD frequency. Behavioral experiments have demonstrated that these "shes detect the phase di!erence between their own EOD and jamming EOD down to 1 ls [4,7,8]. The phase is represented by the timing of zero-crossing of EOD which are represented by impulses of T-type receptor cells. Each small cell in the lamina VI of the TS is sensitive to coincidence of arrival times between the two impulses arising from the T-receptor cells locating at di!erent sites on the "sh body surface. The two-dimensional array of small cells detects the time di!erence with an accuracy of 10 ls [2,3]. The accuracy corresponds to of the usual neural relaxation time (&1 ms). It is remarkable that the system constructed by neurons with much longer processing time can achieve such a short temporal resolution. The neural mechanism realizing the hyperacuity has not yet been made clear. In the present study, we present a neural network model of the torus for the detection of phase advances and delays, which has been made based on the physiological and anatomical data. The network model discriminates phase advance and delay by detecting the time di!erence between a!erent nerve spikes arising from di!erent sites on the body surfaces and integrating the spatio-temporal "ring pattern of the small cells in the lamina during JAR. The single-chain model including a single giant cell axon detects the phase advance and delay up to 60 ls. The detection ability is much increased by integrating the outputs of small cells in the two-dimensional arrays consisting of multiple chains shown in Fig. 1. The discrimination accuracy of the lamina model is noticeably improved by adding noises with optimal intensity to the a!erent signals with a smaller time di!erence. The underlying mechanism is similar to that of stochastic resonance, but the method picking up only meaningful signals from noise-enhanced response spikes is new.
Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862
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Fig. 1. A neural network model of the torus for detection of phase advance and delay.
2. A network model of the torus for detection of phase advances and delays (di4erential phase coder, DPC) Neurons in the torus semicircularis, or small cells, compare timings of a!erent nerve spikes arising from two di!erent parts of the body surface. At early stage of JAR, the phase of EOD wave is encoded as the timing of zero-crossings of EOD wave by T-receptors, and the timings are represented by a!erent nerve spikes. The a!erent signals are converged to giant cell bodies and small cell dendrite in the torus, via spherical cell in the medulla, as shown in Fig. 1 [4]. Each small cell receives two kinds of signals from its own dendrite and from a single giant axon, respectively, which are received signals from di!erent body surface areas as shown in Fig. 1. A small cell detects the coincidence of arrival between the two signals. We proposed a cellular model of a small cell which acts as a coincidence detection unit [6]. We showed that the small cell model detects the coincidence between two input spikes with an accuracy of tens of ls. However, to perform a correct JAR, it is required [4] to detect whether the signal from area A is advanced or delayed compared with that from area B. To detect the phase advances and delays, we presented here a network model which consists of multiple linear arrays of the small cells (coincidence detection (CD) chain) in lamina 6 and two types of neurons, A- and D-neurons in lamina 8b, as shown in Fig. 1. We call hereafter the network model di!erential phase coder (DPC). A CD chain consists of small cells and delay lines (bold line from a giant cell shown in Fig. 1). Each small cell of CD chain receives a spike with time delay depending on the distance between the small cell and the giant cell. The small cell receives simultaneously the spike from the spherical cell, which has a time delay common to all the small cells. Because it has been suggested [2,3] that the torus does not have the ordered map structure with respect to time delay such as the nucleus laminaris of barn owl, the time
858
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of spike propagation between neighboring small cells was set in the model to be di!erent depending on the position of the small cell along the CD chain. The outputs of small cells in each CD chain are integrated by A- and D-neurons of a deeper laminar 8b, as shown in Fig. 1. The mathematical description of a small cell model is given in Ref. [6]. The membrane potentials, < , and the outputs, ; , of A- and D-neurons 7 7 (>"A, D) are determined by 1 ,* , d< 7 "!< # q w z (>"A, D), (1) 7 N 7 GI GI 7 dt * I G 1 z " , (2) GI 1#exp(!(v !h )/e) GI I w "w exp(!(i!1)/p )#a , (3) " GI I " w "w exp(!(N!i)/p )#a , (4) GI I 1 ; " , (5) 7 1#exp(!(< !h )/e ) 7 7 7 where w is the strength of connection from the ith small cell of the kth CD chain to 7 GI >-neuron and decays exponentially depending on the distance between the small cell and >-neuron (>"A, D). z and v are the output and the membrame potential of GI GI the ith neuron in the kth CD chain, respectively, and ; is the "ring rate of neuron >. 7 In the present paper, small cells in each CD chain were arranged linearly with an equal distance for simplicity. The "ring rates of A- and D-neurons increase with increasing the magnitude of phase advance and delay because of the exponential decay of the synaptic strength w . 7 GI 3. Results 3.1. Ability of DPC network to detect phase advances and delays We investigated the detection ability of the DPC model for phase advance and delay in the case where the model has a single CD chain. We used here two types of jamming signals received at two di!erent sites on body surface; one signal whose distortion from its own EOD due to jamming is very small and the other signal whose distortion is large. The former signal is transmitted to the small cell soma, while the latter is transmitted to the small cell dendrite. The spike trains of peripheral nerve arising from these signals were calculated by T-receptor model [5]. The time disparity between two spikes from a giant cell and a spherical cell is changed sinusoidally as shown in Fig. 2A(ii), where the maximum time disparity (¹D ) is 160 ls. Then, the sites of small cell excited the most strongly by the two
signals are changed as the magnitude of disparity is changed, as shown in Fig. 2A(iii).
Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862
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Fig. 2. (A) (i) Jamming signal. (ii) Temporal variation of time disparity detected by T-a!erent model. The time di!erence was measured with reference to the timing of spike from a giant cell. (iii) The spatio-temporal variation of "ring rates of CD units linearly arranged along the vertical axis. The firing rates are plotted in the direction perpendicular to the sheet. This is a two-dimensional projection of the rates. (iv) Temporal variations of "ring rates of A- and D-neurons. (B) The dependence of the di!erence *o"o !o between " o and o on the maximum shift N of the input spikes due to noise. (C) The dependence of the "
ampli"cation ratio r of the di!erence of average "ring rates on ¹D . The `subthresholda and `supra6
thresholda indicate the region of ¹D which is subthreshold and suprathreshold for a clear detection in
the absence of noise, respectively.
The sites of the small cells with the maximum "ring rate move from the delay side to the advance side in the linear array of small cells. The small cells only in a de"nite region of CD chain can "re actively as seen in Fig. 2A(iii), and the number of "ring cells corresponds to the accuracy for detection of time disparity by the CD chain. The accuracy accomplished by the single linear chain of the small cells is about 100 ls, because the time delay between a small cell and its neighbor is set to be 10 ls. Fig. 2A(iv) shows the "ring rates of A- and D-neurons. In the region of phase advance (negative time disparity), the "ring rate of A-neuron is larger than that of D-neuron. On the other hand, in the region of phase delay (positive time disparity), the "ring rate of D-neuron is larger than that of A-neuron. Thus, A- and D-neuron discriminates correctly the phase advance and delay, respectively.
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To evaluate the detection ability of the DPC chain model, we calculated the average "ring rates of A- and D-neurons for the spike patterns with various values of ¹D .
The average "ring rates of A- and D-neurons, o6 and o6 (X"dl, ad) were calculated " in the periods of phase delay (0}100 ms) and phase advance (100}200 ms) of one cycle of the jamming beat pattern. The di!erence between the average "ring rates of A- and D-neurons in the delay region is decreased as ¹D is decreased. For ¹D less than
60 ls, the average "ring rate, o , noticeably decreases. The chain model can discrimi" nate correctly the phase advance and delay for ¹D larger than 60 ls.
3.2. A positive role of input noises in hyperaccurate detection To investigate the role of input noise in the improvement of the detection acuity for the DPC network model with multiple CD chains shown in Fig. 1, we calculated the di!erence in the average "ring rate between A- and D-neurons under the application of input noises with various intensities. The result is shown in Fig. 2B. The interspike intervals into which phase modulation of a jamming signal is encoded are shifted randomly by values within the region from !N to N . As the width of random
shift increases, the di!erence in the average "ring rate between A- and D-neurons increases and then slightly decreases. It becomes maximum at a certain value of the width (N "60 ls). The detection ability of phase advance and delay becomes
higher with increasing of the di!erence of the "ring rate. To evaluate the e!ect of noise on the improvement of detection ability, we calculated the ratio, r "*o6 /*o6, where *o6 (X"ad, dl) is the di!erence in the 6 , , average "ring rate between A- and D-neurons under the application of the optimal noise whose width is N "60 ls, and *o6 is that in the absence of noise. r means
6 the ampli"cation ratio of the detection ability under the application of noise. The result is shown in Fig. 2C. The ratio increases as ¹D decreases. For ¹D smaller
than 60 ls, the ratio noticeably increases, while for ¹D larger than 60 ls, the ratio
remains constant. It appears that the optimal noise enhances more e!ectively the sensitivity of A- and D-neurons for smaller ¹D , which are subthreshold for the
clear detection in the absence of noise. The mechanism of improvement may be similar to that in the stochastic resonace [11]. When the input spike train with the small ¹D (440 ls) does not include the
noise, the "ring region of the CD chain slightly #uctuates temporally within the region of CD layer. The shift is within the subthreshold. When the input spike trains include the noise, the "ring region of the CD chain is shifted more largely to the delay or advance side of the CD chain. The shift is stochastically out of the threshold. Because the magnitude of spike interval shift is chosen randomly from the range from !N to N , the "ring region is shifted more to the delay region in one CD
chain, but more to the advance region in another CD chain. When the noises with optimal width are added to the input spike train with a subthreshold time di!erence, the noises are able to enhance the small shift of "ring zone relevant to the phase advance and delay, because the optimal noise enhances cooperatively the subthreshold shifts over the CD chains.
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References [1] J. Bastian, Electrolocation, Behavior, Anatomy, and Physiology, in: T.H. Bullock, W. Heiligenberg (Eds.), Electroreception, Wiley, New York, 1986, pp. 577}612. [2] C.E. Carr, Processing of temporal information in the brain, Ann. Rev. Neurosci. 16 (1993) 223}243. [3] C.E. Carr, M.A. Friedman, Evolution of time coding systems, Neural Comput. 11 (1999) 1}20. [4] W. Heiligenberg, Neural Nets in Electric Fish, Chaps. 3 and 4, MIT Press, Cambridge, 1991. [5] Y. Kashimori, M. Goto, T. Kambara, A Model of P- and T-electroreceptors of weakly electric "sh, Biophys. J. 70 (1996) 2513}2526. [6] Y. Kashimori, T. Kambara, A neural mechanism for hyperacuity in jamming avoidance response of electric "sh Eigenmannia, Proceedings of 1997 IEEE International Conference on Neural Networks Vol. 1, 1997, pp. 332}337. [7] M. Kawasaki, G. Rose, W. Heiligenberg, Temporal hyperacuity in single neuron of electric "sh, Nature (London) 336 (1989) 173}176. [8] M. Kawasaki, Sensory hyperacuity in the jamming avoidance response of weakly electric "sh, Curr. Opin. Neurobiol. 7 (1997) 473}479. [9] M. Konishi, Listening with two ears, Sci. Am. 268 (1993) 66}73. [10] J.A. Simmons, The resolution of target range by echolocating bat, J. Acoust. Soc. Am. 54 (1973) 157}173. [11] K. Wiesenfeld, F. Moss, Stochastic resonance and the bene"ts of noise: from ice ages to Cray"sh and Squids, Nature 373 (1995) 33}36.
Yoshiki Kashimori received his Ph.D. from Osaka City University in 1985. He is a research associate in the Department of Applied Physics and Chemistry, at the University of Electro-Communications. His research interest is to clarify the neural mechanism of information processing in the electrosensory, auditory, and gustatory systems, based on the modeling of neurons and their network. He also investigates the emergence of dynamical orders in various biological systems, based on the nonlinear dynamics.
Osamu Hoshino received his Ph.D. in Biophysics from the University of Electro-Communications in 1998. He is presently an assistant professor in the Department of Human Welfare Engineering at Oita University. His research interests include the neural mechanisms of olfactory, auditory, and memory systems. He is also interested in the neural basis of cortical mapping and its role in human behavior.
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Y. Kashimori et al. / Neurocomputing 32}33 (2000) 855}862 Takeshi Kambara received his Ph.D. from Tokyo Institute of Technology in 1970. He is a Professor of Biophysics in the Department of Applied Physics and Chemistry and a Professor of biological information science in the Graduate School of Information Systems, at the University of Electro-Communications. His scienti"c interests cover the neural mechanism of information processing in olfactory, auditory, visual, gustatory, and electro-sensory systems, the brain model of consciouness, and emergence of dynamical orders in various biological complex systems. His research work uses the `in silicoa method.