Lecture Notes in Computer Science
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Frequency(Hz) M1 20.50±1.92 21.80±1.98 22.48±2.04 23.73±1.84 18.57±1.2 20.75±1.62 18.23±1.82 S1 21.17±1.79 21.80±1.97 23.08±1.82 23.10±2.15 18.22±2.13 21.17±1.08 19.52±.3.21
between 19 EEG channels and EMG (FDI) for all subjects (n=7) are superimposed and topographically according to the approximate locations of the electrodes on the scalp. Each electrode labeled with respect to its location (Fp: Frontal pole F: Frontal T: Temporal C: Central P: Parietal O: Occipital). B, Expanded view of the coherence spectra between EEG (C3 scalp site) and EMG (FDI) for all subjects (n=7). Each line style specifies different subject’s coherence spectra. Coherence values only above 99% significance level (indicated by the solid horizontal line) are highlighted.
The Effects of Theta Burst Transcranial Magnetic Stimulation 139 band at C3 scalp site. Maximum coherence levels were observed at C3 (n=6) and at F3 (n=1) scalp sites whereas no significant coherence was observed at other locations (Figure 2). These results are in well agreement with the previous studies on coherence between EEG-EMG during isometric contraction [7, 8]. Table 1 shows the average absolute coherence values and peak frequencies at all trials before and after the application of TBS. Figure 3 demonstrates the normalized EEG (C3)-EMG coherence values as average of all subjects. Coherence values obtained before the TBS were taken as control and set to 100%. Average beta band coherence suppressed to 56.2% after 30 minutes and 54.5% after 60 minutes with statistical significance(p<0.05, Bonferonni multiple comparison test) and recovered to the original control level 90 and 120 minutes when TBS applied over M1. However there is no statistically significant coherence change with respect to time when TBS is delivered to S1 (coherence values are 103.8% after 30 minutes, 99.8% after 60 minutes).
taken as control level (100%). Coherence was suppressed (* p<0.05, Bonferonni multiple comparison test) 30 and 60 minutes after TBS-over-M1 and recovered back to the control level. Error bars indicate standard error of mean (±SEM).
The present results show that TBS over human motor cortex can change the cortico- muscular functional coupling. TBS-over-M1 inhibited EEG (C3)-EMG (FDI) coherence for about 60 minutes, this inhibition has similar time course with MEP 140 M. Saglam et al. amplitude suppression after TBS-over-M1 [4, 5]. The parallel responses of corticomuscular coherence and MEP to TBS may suggest that TBS-driven LTP/LTD of synaptic connections, involved in the circuits for MEP generation (i.e. I wave circuits), not only produces MEP but also plays an important role in rhythmic activities in the motor system. However it is difficult to assess the exact LTP/LTD scenario after TBS in conscious human experiments. On the other hand, we know that effects of cTBS count on pre- and postsynaptic N-methyl-D-aspartate receptors (NMDARs) which are highly associated with LTP/LTD [10, 11]. Blocking NMDARs with pharmacological agents (i.e. memantine) eliminates the inhibitory effects of cTBS [11]. Thus we may conclude that NMDAR-related LTP/LTD of synaptic connections within motor circuit is involved in corticomuscular coherence phenomenon and it can be temporarily modified by TBS. Among conventional rTMS studies, there are few reports investigating the effects of rTMS on cortico-muscular coherence. Similar changes (suppression) in cortico- muscular coherence were observed using longer rTMS conditioning (15min with 0.9Hz rTMS vs. 40s with TBS-over-M1) although it was applied over premotor cortex in that study [9]. Absence of evidence in the literature hinders us to make a complete comparison between rTMS-over-M1 and TBS-over-M1. However it is evident that TBS has prolonged suppressive effect (15min effect with 0.9Hz rTMS vs. 60min effect with TBS). Therefore present data confirms that TBS could be a more prominent technique than rTMS by means of cortico-muscular coherence. Since rTMS has a therapeutic potential in clinical use [12, 13], a shorter yet stronger stimulation technique should be considered. On the other hand, TBS-over-S1 showed no significant coherence change although stimulation site is just 2 cm posterior to M1. The lack of suppressive effect after TBS-over-S1 suggests that there is no conditioning on M1 due to a current spread from S1 to M1 so that TBS can be regarded as a focal stimulation tool. The coherence was suppressed after of TBS-over-M1 rather than TBS-over-S1 condition. This result agrees with the fact that MEP amplitude is suppressed by TBS-over-M1 but not by TBS-over-S1. Therefore present findings could indicate that similar mechanism is evident for the MEP generation and coupling between cortical and muscular sites.
1. Brown, P., Marsden, J.F.: Cortical network resonance and motor activity in humans. Neuroscientist 7(6), 518–527 (2001) 2. Chouinard, P.A., Paus, T.: The primary motor and premotor areas of the human cerebral cortex. Neuroscientist 12(2), 143–152 (2006) 3. Filipovic, S.R., Siebner, H.R., Rowe, J.B., Cordivari, C., Gerschlager, W., Rothwell, J.C., Frackowiak, R.S., Bhatia, K.P.: Modulation of cortical activity by repetitive transcranial magnetic stimulation (rTMS): a review of functional imaging studies and the potential use in dystonia. Adv. Neurol. 94, 45–52 (2004) 4. Huang, Y.Z., Edwards, M.J., Rounis, E., Bhatia, K.P., Rothwell, J.C.: Theta burst stimulation of the human motor cortex. Neuron 45, 201–206 (2005) 5. Ishikawa, S., Matsunaga, K., Nakanishi, R., Kawahira, K., Murayama, N., Tsuji, S., Huang, Y.Z., Rothwell, J.C.: Effect of theta burst stimulation over the human sensorimotor cortex on motor and somatosensory evoked potentials. Clinical Neurophysiology (in press, 2007) The Effects of Theta Burst Transcranial Magnetic Stimulation 141 6. Mima, T., Hallett, M.: Electroencephalographic analysis of cortico-muscular coherence: reference effect, volume conduction and generator mechanism. Clinical Neurophysiology 110, 1892–1899 (1999) 7. Murayama, N., Lin, Y.Y., Salenius, S., Hari, R.: Oscillatory interaction between human motor cortex and trunk muscles during isometric contraction. Neuroimage 14, 1206–1213 (2001) 8. Conway, B.A., Halliday, D.M., Farmer, S.F., Shahani, U., Maas, P., Weir, A.I., Rosenberg, J.R.: Synchronization between motor cortex and spinal motoneuronal pool during the performance of a maintained motor task in man. J. Physiol. 489(3), 917–924 (1995) 9. Chen, W.H., Mima, T., Siebner, T., Oga, H.R., Hara, T., Satow, H., Begum, T., Shibasaki, H.: Low-frequency rTMS over lateral premotor cortex induces lasting changes in regional activation and functional coupling of cortical motor areas. Clinical Neurophysiology 114(9), 1628–1637 (2003) 10. Huang, Y.Z., Chen, R.S., Rothwell, J.C., Wen, H.Y.: The after-effect of human theta burst stimulation is NMDA receptor dependent. Clin. Neurophysiol. 118(5), 1028–1032 (2007) 11. MacDermott, A.B., Mayer, M.L., Westbrook, G.L., Smith, S.J., Barker, J.L.: NMDA- receptor activation increases cytoplasmic calcium concentration in cultured spinal cord neurones. Nature 321(6069), 519–522 (1986) 12. Lefaucheur, J.P.: Repetitive transcranial magnetic stimulation (rTMS): insights into the treatment of Parkinson’s disease by cortical stimulation. Neurophysiol. Clin. 36(3), 125– 133 (2006) 13. Joo, E.Y., Han, S.J., Chung, S.H., Cho, J.W., Seo, D.W., Hong, S.B.: Antiepileptic effects of low-frequency repetitive transcranial magnetic stimulation by different stimulation durations and locations. Clin Neurophysiol 118(3), 702–708 (2007)
Interactions between Spike-Timing-Dependent Plasticity and Phase Response Curve Lead to Wireless Clustering Hideyuki Cˆ ateau 1
2 , and Tomoki Fukai 1 1
2 Department of Human and Computer Intelligence, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan Abstract. A phase response curve characterizes the signal transduction between neurons in a minimal manner,whereas spike-timing-dependent plasiticity (STDP) characterizes the way to rewire networks in an activity- dependent manner. The present paper demonstrates that these two key properties both related to spikes work synergetically to carve function- ally useful circuits in the brain. STDP working on a population of neu- rons that prefer asynchrony turns out to convert the initial asynchronous firing to clustered firing with synchrony within a cluster. They get syn- chronized within a cluster despite their preference to asynchrony because STDP selectively disrupts intra-cluster connections, which we call wire- less eclustering. 1 Introduction Synchrony tendency of coupled oscillators or neurons is predicted by the phase response curve (PRC) of a neuron that describes an amount of advance or de- lay by synaptic input given at a specific phase in an interspike interval[1,2]. It is intriguing to know how this useful theory based on the fixed coupling strength between neurons generalizes to the cases where synaptic strength varies as observed in the real brain. A number of experiments[3] have established that synaptic strength changes depending on pre- and postsynaptic spike times and theoretical implications of such spike-timing-dependent plasticity (STDP) have been extensively studied [4,5,6,7,8,9]. Since the PRC and STDP both refer to the timings of spikes, a natural question is how these two properties of a neuronal network interact each other to carve a functional network in the brain. To answer this question, we first use a neuron model whose PRC can be systematically controlled [10] unlike the simpler leaky integrate-and-fire (LIF) model. The model neurons favors either asynchronous (Model A) or synchronous (Model B) firing depending on the values of the model parameters. Our simu- lations show that STDP working on the network of Model A neurons converts an asynchronously firing neurons into three or more cyclically activated clus- ters of neurons. Model A neurons can synchronize within a cluster despite their preference to asynchrony because, as we see later, STDP selectively disrupts M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 142–150, 2008. c Springer-Verlag Berlin Heidelberg 2008
Interactions between STDP and PRC Lead to Wireless Clustering 143
intra-cluster connections, nullifying the asynchrony preference. If STDP works on the network of Model B neurons, however, the neurons simply get synchro- nized globally, analogous to what was observed in [11], and nothing interesting happens.
Next we show that the self-organized cyclic activity appears also under biolog- ically realistic settings using a Hodgkin-Huxley type neuron model, suggesting the generality of the concept. In the self-organization, PRC specifies the timing preference and influences the way how STDP works. Importantly, STDP in turn influences the way how PRC is readout. Before the STDP learning begins, the initial slope of an effective PRC (defined later) determines the stability of the global synchrony. After the STDP learning forms the cyclic activity consisting of n clusters, the slope of the effective PRC at θ = 2π(1 − 1/n) determines its stability. Thus, the two key features of spiking neurons, PRC and STDP, work synergetically to organize functional networks in the brain. Previously studies have shown [12,13,14] that STDP helps the pacemaker neuron entrain an innervated neuron(s), which was called ”frequency synchrony” meaning that neurons start firing at the same frequency but with different phases as opposed to the ”phase synchrony” studied here. Building upon the firm the- oretical analysis of frequency synchrony [1,12,13,14], it is now important to ask when and how the frequency synchrony specializes to the phase synchrony be- cause the phase synchrony can effectively enlarge otherwise tiny EPSC to the size capable of reliably eliciting spikes in innervated neurons. 2 Self-organization of Izhikevich Neurons We consider a population of neurons firing quasi-periodically. We use a spik- ing neuron model proposed by Izhikevich[10]. Depending on the values of four parameters, a, b, c and d, this model can produce many different voltage tra- jectories similar to what are found in real neurons. Fifty Model A neurons that favor asynchrony (a = 0.02, b = 0.2, c = −50, d = 1.26) are connected in an all-to-all manner with uniform synaptic strength and with a range of synaptic delays of 2 ±0.2ms. The neurons fire quasi-periodically driven by supra-threshold stochastic inputs, I = I 0 + σζ(t) with I 0 = 30mV/ms and σ = 1.5mV/ms 1/2 ,
With no synaptic plasticity at work, initial uniform distribution of firing phases (Fig.1a) remain asynchronous because the neurons favor synchrony. How- ever, the effects of the standard additive STDP rule with hard boundaries (0 < w < 1 ) defined [4] by Δw = A
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20 25 30 35 40 45 50 1 2 1 2 θ θ θ θ Fig. 1. Clustering of Model A neurons by STDP. Fifty Model A neurons fire in com- plete asynchrony before the STDP learning starts. (a) Voltage trajectories of all the neu- rons drawn in different colors are overlaid. (b) The neurons start firing in three different clusters with intra-cluster synchrony due to STDP. (c) A gray scale representation of the connection strength between neurons with black being the strongest. (d) A raster plot corresponding to (b). Neurons are aligned according to spike times. (e) The con- nection matrix when the higher-order rule of STDP is applied. A synaptic change is dis- counted by 1 − exp(−(t post spike 2 − t EPSP by pre ) /τ remove ) with when a triplet of event, t post spike 1 < t EPSP by pre < t post spike 2 , happened. (f) A schematic drawing showing the network topology corresponding to (d) and (e). (g), (h) Positive feedback mecha- nism leading to the wireless clustering. In (g), the vertical lines indicate spike times of two neurons; the small wedges indiate EPSPs elicited by spikes. In (h), distributed firing phases of neurons are represented by the filled circules in different colors. The network topology underlying this clustered synchronous is picturized by a matrix of synaptic strengths (Fig.1c), where neurons are indexed according to firing times after the learning (t > 9.89sec). The three divisions apparent in Fig.1c correspond to three synchronously firing clusters of neurons (Fig.1d).
Interactions between STDP and PRC Lead to Wireless Clustering 145
Interestingly, STDP appears to have removed the intra-cluster connections al- most completely (Fig.1d). Such clustering without connections is observed com- monly under various simulation conditions and we will call it wireless clustering. It turns out the wireles clustering has happened because we reflected experi- mental procedures of STDP faithfully. The standard STDP rule implies poten- tiation for positive timing difference, Δt = t post − t
pre > 0, and depression for a negative timing difference, Δt < 0. Many have argued that the asymmetry of this rule produces a one-way coupling (see e.g. [4] ). Such arguments would be valid if Δt represented the time difference between post- and presynaptic spikes. However, actually most experimental literatures [3] define Δt to be the time difference between a postsynaptic spike and the onset or peak of the so- matic excitatory postsynaptic potential (EPSP) induced by a presynaptic spike: Δt = t post spike − t EPSP by pre . Hence the above argument does not apply. A so- matic EPSP should lag behind a presynaptic spike for a few msec. Therefore, if two neurons fire in exact synchrony (Fig.1g), Δt < 0 negative [12] for both directions, thereby weakens connections bidirectionally. Now, how does this mechanism convert initial asynchronous firing to clustered synchronous firing (Fig.1b)? Initial asynchronous firing (Fig.1a) is represented as firing phases evenly spread around the circle (Fig.1h, left). The firing re- mains asynchronous without STDP . However, with the phases of many neurons squeezed into the circle, any single neuron must have neighboring neurons that unwillingly fire synchronously with it(Fig.1h). Among these neurons, the above- mentioned mechanism weakens the connections bidirectionally. As their synap- tic connections weaken, mutual repulsion is also weakened. This then further synchronizes their firing. This positive feedback mechanism develops wireless clusters (Figs.1h). Although this mechanism qualitatively explains how the clus- tering happens, a quantitative question how many clusters are formed requires further consideration. We will later see that a stability analysis tells the possible number of clusters. In contrast to the vanishing intra-cluster connections, the inter-cluster con- nections survive and can be unidirectional (Fig.1d), which defines the cyclic network topology such as shown in Fig.1f, upper. Let us ask how we can change this 3-cycle topology. We find that one of the recently observed higher-order rules of STDP [15,16] increases the number of clusters (Fig.1e). The higher-order rule shown in [16] implies the gross reduction in the LTD effect because LTP override the immediately preceding LTD, while LTP simply cancels partly the immediately preceding LTD. The weakened LTD effect is likely to increase the total number of potentiated synapses, which is in consistent with the increased ratio of black areas in Fig.1e compared to Fig.1d. In contrast to such cluster-wise synchrony observed with Model A neurons, Model B neurons that favor synchrony ( a = 0.02, b = 0.2, c = −50, d = 40) self- organize into the globally synchronous state with or without synchrony (Fig.2). Due to the global synchrony, mutual synaptic connections are largely lost, and each neuron ends up being driven by the external input individually, having little sense of being present as a population. The global synchrony gives too strong
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10000 0 5 10 15 20 25 30 35 40 45 50 time [ms] 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 neuron index presynaptic neurons postsynaptic neurons Fig. 2. Global synchrony observed with Model B neurons that favor synchrony. A raster plot (a) and connection matrix (b) of fifty Model B neurons. The neurons were aligned with the connection-based method: neurons are defined to belong to the same cluster whenever there mutual connections are small enough. an impact and also has minimal coding capacity because all the neurons behave identically, and it appears to bear more similarity to the pathological activity such as seizure in the brain than to meaningful information processing. By contrast, the clustered synchrony arising in the network of Model A neu- rons appears functionally useful. Generally in the brain, the unitary EPSP am- plitude ( ∼ 0.5mV ) is designed to be much smaller than the voltage rise needed to elicit firing ( ∼ 15mV ). Therefore, single-neuron activity alone cannot cause other neurons to respond. Hence, it is difficult to regard the single-neuron activ- ity as a carrier of information transferred back and forth in the brain. In contrast, the self-organized assembly of tens of Model A neurons (Figs. 1d) looks an ideal candidate for a carrier of information in the brain because their impact on other neurons are strong enough to elicit responses. Additionally, a cluster can reliably code the timing information. The PRC, Z(2πt/T ), representing the amount of advance/delay of the next fir- ing time in response to the input at t in the firing interval [0, T ] has been mostly used to decide whether a coupled pair of neurons or oscillators tend to syn- chronize or desynchronize under the assumption that the connection strengths between the neurons are equal and unchanged. Specifically, suppose that a pair of neurons are mutually connected and a spike of one neuron introduces a cur- rent with the waveform of EP SC(t) in an innervated neuron after a transmission delay of τ d . The effective PRC defined as Γ − (θ) =
1 T T 0 Z(2πt /T )EP SC(t − τ d
θ 2π )dt is known to decide their synchrony tendency. If Γ − (θ) < 0 at θ = 0 is positive (negative), the two neurons are desynchronized (synchronized). This synchrony condition is inherited to a population of neurons coupled in an all- to-all or random manner as far as the connection strengths remain unchanged. Theoretically calculated Γ − (θ)s for Model A and B (Fig.3a,b) explain that the all-to-all netwok of Model A (B) neurons exhibit global asynchrony (synchrony). Note that both Model A and B neurons belong to type II[19] so that both model neurons favor synchrony if they are delta-coupled with no synaptic delay. After STDP is switched on, the network consisting of Model A neurons, is self- organized into the 3-cycle circuit (Fig.1d) with a successive phase difference of the clusterd activity being Δsucθ = 2π/3. Stability analysis shows that the slope Interactions between STDP and PRC Lead to Wireless Clustering 147
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2pi/4 Fig. 3. Effective PRCs and schema of triad mechanism. The effective PRC of Model A and B were calculated with the adjoint method [19] and shown shown in (a) and (b). The slope at θ = 0 is positive for (a) but negative for (b) although it hardly recognizable with this resolution. The slope at θ = 2π/3 is negative for (a) but positive for (b). The dashed lines represent θ = 2π/3 and θ = 2π/4. of Γ
− (θ) not at the origin but at θ = 2π − Δsucθ now determines the stability of the 3-cycle activity: the 3-cycle activity is stable if Γ − (2π − Δsucθ) < 0. Fig.3a tells that the 3-cycle activity shown in Fig.1c is stable. The stable cyclic activity is achieved through the following synergetic process: (1) PRC determines the preferred network activity (e.g. asynchronous or syn- chronous), (2) the network activity determines how STDP works, STDP modifies the network structure (e.g. from all-to-all to cyclic ), and (3) the network struc- ture determines how the PRC is readout (e.g. θ = 0 or θ = 2π −Δsucθ) ), closing the loop. Generally, we can show that the n-cylce activity whose successive phase differ- ence equals Δsucθ = 2π/n is stable if Γ − (2π − Δsucθ) < 0. PRCs of biologically plausible neuron models or real neurons [20] tend to have a negative slope in a later phase of the firing interval and converge to zero at θ = 2π because the mem- brane potential starts the regenerative depolarization and becomes insensitive to any synaptic input. The corresponding effective PRCs inherit this negative slope in the later phase and tends to stabilize the n-cycle activity for some n. 3 Self-organization of Hodgkin-Huxley Type Neurons Next we see that the self-organized cyclic activity with the wireless clustering is also observed in biologically realistic setting. Our simulations as described in [18] with 200 excitatory and 50 inhibitory neurons modeled with the Hodgkin- Huxley (HH) formalism exhibits the 3-cyclic activity with the wireless clustering (Fig.4a,b). The setup here is biologically realistic in that (1) HH type neurons are used, (2) physiologically known percentage of inhibitory neurons with non- plastic synapses are included, (3) neurons fire with high irregularity due to large noise in the background input unlike the well-regulated firing as shown in Fig.1c. Interestingly, the effective PRC (Fig.4c) of the HH type neuron shares impor- tant features with that of Model A: the positive initial slope implying the pref- erence to asynchrony and a negative later slope stabilizing the 3-cycle activity.
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time(msec) neuron index pi Fig. 4. Conductance-based model also develops the wireless clustering. (a) A raster plot of 200 HH-type excitatory neurons showing 3-cycle activity. (b) The correspond- ing connection matrix showing the wirelessness. (c) Effective PRC or Γ −
θ) of the conductance-based model. Generally, technical difficulty in the HH simulations is their massive computa- tional demands due to the complexity of the system. That difficulty has hidered theoretical analysis, and has left the studies largely experimental. In particular, previously we tried hard to understand why we never observed 4-cycle or longer in vain. However, the analytic argument we developed here with the simplified model gives a clear insight into the biologically plausible but complex system. Comparison of Fig.3a and Fig.4c reveals that the negative slope of Γ − (θ) of the HH model is located at more left than that of Model A, indicating less stability of long cycles in the HH simulations. With the larger amount of noise in the HH simulations in mind, it is now understood that 4-cycle and longer can be easily destabilized in the HH simulations. Thus, our analysis developed with the simplefied system serves as a useful tool to understand a biologically realistic but complex systems. There is, howevr, an interesting difference between the Model A and HH simulations. Although the intra-cluster wirelessness is a fairly good first approx- imation in the HH model simulations (Fig.4b), it is not as exact as in the Model A simulations (Fig.1d,e). Interestingly, an elimination of the residual intraclus- ter connections destroys the cyclic activity, suggesting the supportive role of the tiny residual intra-cluster connections. 4 Discussion In the previous simulation study[17] using the LIF model, cyclic activity was ob- served to propagate only at the theoretical speed limit: it takes only τ d from one
cluster to the next, requiring the zero membrane integration time. To understand Interactions between STDP and PRC Lead to Wireless Clustering 149
why it was the case, we first remind that the effective PRC needs a negative slope at 2π − Δ suc θ to stabilize the cylic activity. However, the slope of the PRC of an LIF model, Z(θ) = c exp( T τ m θ 2π ), is always positive except at the the end point, where Z(2π − 0) = c exp( T τ
) and Z(2π + 0) = c, implying Z (2π) = −∞. This infinitely sharp negative slope of the PRC at θ = 2π is rounded and displaced to 2π − 2πτ d /T in Γ − (θ) (see its definition). Since this is the only place where Γ −
suc θ = 2πτ
d /T ,
implying the propagation at the theoretical speed limit. We demonstrated an intimate interplay between PRC and STDP using the Izhikevich neuron model as well as the HH type model. The present study com- plements previous studies using the phase oscillator [11,14], where its mathemati- cal tractability was exploited to analytically investigate the stability of the global phase/frequency synchrony. The self-organization or unsupervised learning by STDP studied here complements the supervised learning studied in [22]. The propagation of synchronous firing and temporal evolution of synaptic strength under STDP is know to be analyzed semi-analytically with the Fokker-Planck equation STDP [5,6,8,9,21]. It is interesting future direction to see how the Fokker-Planck equation can be used to understand the interplay between PRC and STDP. Acknowledgement The present authors thank Dr. T. Takewaka at RIKEN BSI for offering the code to calculate the PRC. References 1. Kuramoto, Y.: Chemical oscillations,waves,and turbulence. Springer, Berlin (1984) 2. Ermentrout, G., Kopell, N.: SIAM J. Math.Anal. 15, 215 (1984) 3. Markram, H., et al.: Science 275, 213 (1997); Bell, C.C., et al.: Nature 387, 278 (1997); Magee, J.C., Johnston, D.: Science 275, 209 (1997); Bi, G.-Q., Poo, M.-M.: J. Neurosci. 18, 10464 (1998); Feldman, D. E., Neuron 27, 45 (2000); Nishiyama, M., et al.: Nature 408, 584 (2000) 4. Song, S., et al.: Nat. Neurosci. 3, 919 (2000) 5. van Rossum, M.C., Turrigiano, G.G., Nelson, S.B.: J. Neurosci. 22,1956 (2000) 6. Rubin, J., et al.: Phys. Rev. Lett. 86, 364 (2001) 7. Abbott, L.F., Nelson, S.B.: Nat. Neurosci. 3, 1178 (2000) 8. Gerstner, W., Kistler, W.M.: Spiking neuron model. Cambridge University Press, Cambridge (2002) 9. Cˆ ateau, H., Fukai, T.: Neural Comput., 15, 597 (2003) 10. Izhikevich, E.M.: IEEE Trans. Neural Netw. 15, 1063 (2004) 11. Karbowski, J.J., Ermentrout, G.B.: Phys. Rev. E. 65, 031902 (2002) 12. Nowotny, T., et al.: J. Neurosci. 23, 9776 (2003) 13. Zhigulin, V.P., et al.: Phys. Rev. E, 67, 021901 (2003) 14. Masuda, N., Kori, H.: J. Comp. Neurosci, 22, 327 (2007) 15. Froemke, R.C., Dan, Y.: Nature, 416, 433 (2002) 150 H. Cˆ
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ateau, H., Reyes, A.D.: Phys. Rev. Lett. 96, 058101, and references therein (2006) 22. Lengyel, M., et al.: Nat. Neurosci. 8, 1677 (2005) M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 151–159, 2008. © Springer-Verlag Berlin Heidelberg 2008 Download 12.42 Mb. Do'stlaringiz bilan baham: 
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