Hoppensteadt F.C. and Izhikevich E.M. (1996b) Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning of phase information. Biological Cybernetics, 75:129--135.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 51 s 52 s 53 s 54 PLL Neural Network I(t) V( q q=W w( w( t I(t) V( q V( q Figure 6: Conceptual architecture of the phase locked loop neural network described by (12) Notice that the voltage controlled oscillator (VCO) outputs are phase shifted by Gamma90 o ( Gamma=2) modified from Hoppensteadt and Izhikevich 2000b) There is an interesting but still open question regarding the memory capacity of the oscillatory neural networks. Abbot (1990) used H ij ( sin to show numerically that the capacity is comparable with that of the Hopfield network. 5 Possible Hardware Implementations The major advantage of ....

....this section we discuss such a possibility. Phase Locked Loop Neurocomputers. Dynamical system of the form # i = Omega V (# i ) n X j=1 s ij V (# j Gamma =2) 12) describes oscillatory network consisting of phase locked loop (PLL) circuits depicted in Fig. 6 (the details are provided by Hoppensteadt and Izhikevich 2000b) Here # i is the phase of the voltage controlled oscillator (VCO) embedded in the ith PLL, Omega AE 1 is its natural frequency (MHz) s ij are connection coefficients, and V is a 2 periodic output waveform function. It is assumed to satisfy certain odd even condition. Three out of four ....

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Hoppensteadt F.C. and Izhikevich E.M. (1996b) Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning of phase information. Biological Cybernetics, 75:129--135.

....negative, the rest state becomes unstable, and an oscillation builds up. From a dynamical systems point of view this corresponds to an Andronov Hopf bifurcation. To keep our exposition as simple as possible, we do not provide a detailed derivation of the canonical model, which is done in [8] 9] [10]. In Sect. IV we apply the theory to pattern recognition problem. II. A Single MEMS Oscillator. MEMS oscillators are being developed to provide miniature substitutes for crystal oscillators in wireless communication and signal processing applications [15] Detailed information about their ....

....describe the strength of electrical and mechanical connections from the jth to the ith oscillator, respectively. If the oscillators are coupled via soft springs, then (10) needs additional equations for spring dynamics. System (10) has been investigated in Chapters 5 and 10 in [8] and in [9] [10]. There one can find details of the reduction of (10) to the canonical model z i = c i i )z i (a ib)z i jz i j 2 n X j=1 c ij z j ; 11) where z i = x i i x i , c ij = 1 2m p ij Gamma i 2m k ij ; 12) and the other parameters were defined in the previous section. A. ....

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Hoppensteadt F.C. and Izhikevich E.M. Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning of phase information, Biological Cybernetics, 75:129-- 135, 1996

....negative, the rest state becomes unstable, and an oscillation builds up. From a dynamical systems point of view this corresponds to an Andronov Hopf bifurcation. To keep our exposition as simple as possible, we do not provide a detailed derivation of the canonical model, which is done in [8] [9], 10] In Sect. IV we apply the theory to pattern recognition problem. II. A Single MEMS Oscillator. MEMS oscillators are being developed to provide miniature substitutes for crystal oscillators in wireless communication and signal processing applications [15] Detailed information about their ....

....which describe the strength of electrical and mechanical connections from the jth to the ith oscillator, respectively. If the oscillators are coupled via soft springs, then (10) needs additional equations for spring dynamics. System (10) has been investigated in Chapters 5 and 10 in [8] and in [9], 10] There one can find details of the reduction of (10) to the canonical model z i = c i i )z i (a ib)z i jz i j 2 n X j=1 c ij z j ; 11) where z i = x i i x i , c ij = 1 2m p ij Gamma i 2m k ij ; 12) and the other parameters were defined in the previous section. ....

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Hoppensteadt F.C. and Izhikevich E.M. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model, Biological Cybernetics, 75:117--127, 1996

....argument (angle) difference between the complex numbers k i and k j . For example, the merry go around state can be represented as = 1; e i=n ; e i2=n ; e i3=n ; e i(n Gamma1) n ) 2 C n : To memorize such phase patterns we can employ the complex Hebbian learning rule [8] [9], 7] c ij = 1 n p X k=0 k i k j ; which modifies both the strength and the phase of the synaptic connections. The generalized PLL neural network (8) has more than one implementation. A straightforward one is when each circuit connecting two neurons has modifiable weight, s ij , and ....

Hoppensteadt F.C. and Izhikevich E.M. Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning of phase information, Biological Cybernetics, 75:129-- 135, 1996

....the argument (angle) difference between the complex numbers k i and k j . For example, the merry go around state can be represented as = 1; e i=n ; e i2=n ; e i3=n ; e i(n Gamma1) n ) 2 C n : To memorize such phase patterns we can employ the complex Hebbian learning rule [8], 9] 7] c ij = 1 n p X k=0 k i k j ; which modifies both the strength and the phase of the synaptic connections. The generalized PLL neural network (8) has more than one implementation. A straightforward one is when each circuit connecting two neurons has modifiable weight, s ij , ....

Hoppensteadt F.C. and Izhikevich E.M. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model, Biological Cybernetics, 75:117--127, 1996

....this type of bifurcation, and in this sense it is closely related to the VCON. We describe here some interesting aspects of wave propagation in networks that are near multiple SNLCs by studying networks of VCONs. This work is related to modeling regions of the neocortex and other brain structures [5,7], and these models are similar to many others that have been derived for parts of the neocortex using neurooscillators [8,15] An advantage of the VCON model is that it enables one to study information flow in networks using routine frequency domain methods. A VCON network is described by the ....

F. C. Hoppensteadt, E. M. Izhikevich, Synaptic Organizations and dynamical properties of weakly connected neural oscillators, I. Analysis of a canonical model, II. Learning Phase Information, Biological Cybernetics, 75(1996)117-127, 129-135.

....while the delays are either kept constant or neglected. In Section VII we present a (complex conjugate Hebbian) learning rule that changes the delays to memorize temporal patterns. The learning rule was derived originally for weakly connected oscillators near multiple Andronov Hopf bifurcation [16], and it seems to work for pulse coupled oscillatory networks too. II. The Phase Model. A. General Pulse Coupled Neural Networks. Many pulse coupled networks can be written in the following form x i = f i (x i ) n X j=1 g ij (x i )ffi(t Gamma t j Gamma j ij ) 1) Here x i 2 [0; ....

....C = c ij ) is self adjoint; i.e. c ij = c ji for all i and j. Notice, that this condition is general and does not depend on form of the odd function g. The requirement that C is self adjoint arises naturally in weakly connected networks near multiple Andronov Hopf bifurcations [14] 15] [16], for which a learning rule is wellknown: Let the complex variable z i = e 2 ix i denote the periodic activity of the ith neuron (The neuron fires when z i crosses the positive part of the real line R. The learning rule [14] 16] has the following simple form c ij = ffz i z j Gamma flc ij ....

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Hoppensteadt F.C. and Izhikevich E.M. Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning of phase information, Biological Cybernetics, 75:129-- 135, 1996

....matrix C = c ij ) is self adjoint; i.e. c ij = c ji for all i and j. Notice, that this condition is general and does not depend on form of the odd function g. The requirement that C is self adjoint arises naturally in weakly connected networks near multiple Andronov Hopf bifurcations [14] [15], 16] for which a learning rule is wellknown: Let the complex variable z i = e 2 ix i denote the periodic activity of the ith neuron (The neuron fires when z i crosses the positive part of the real line R. The learning rule [14] 16] has the following simple form c ij = ffz i z j Gamma ....

Hoppensteadt F.C. and Izhikevich E.M. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model, Biological Cybernetics, 75:117--127, 1996

....of such networks and their dynamical properties. In particular, there are some general restrictions on the values of phase deviations that can be observed in such networks. Another severe restriction exists if an oscillatory neural network is to memorize and reproduce arbitrary phase deviations [4] using local Hebbian learning rule. One of the requirements is that the excitatory neurons have long axons capable of forming synaptic connections with distant excitatory and inhibitory neurons. The inhibitory neurons are allowed to have short axons and function as local circuit neurons or to have ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: II. Learning of Phase Information. Biological Cybernetics, 75:129--135, 1996

....One of the important questions to ask is: What certain brain structures cannot accomplish no matter what the equations are Thus, studying canonical models can reveal some general laws and restrictions. Our preliminary results shows that such restrictions exist. For example, when we studied [3] weakly connected networks of neural oscillators, we discovered that there is a relationship between synaptic organizations of such networks and their dynamical properties. In particular, there are some general restrictions on the values of phase deviations that can be observed in such networks. ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model. Biological Cybernetics, 75:117--127, 1996

....multiple supercritical Andronov Hopf bifurcation. Both cases are analyzed in great detail in [3] 7. Discussion Observe that our analysis determines whether any interactions between local populations of neurons are possible or not. It does not tell anything about the interactions. We hypothesize [2] that they are based on phase deviations. There is a temptation to identify the mechanism of linking and separating cortical columns that we described above with the Dynamic Link Architecture (von der Malsburg [4] but there is a difference between them. The latter is based on phase deviations ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: II. Learning of Phase Information. Biological Cybernetics, 75:129--135,

....i = Omega j , then it is easy to see that e ij = 0, but c ij is usually non zero, provided there are synaptic connections from x j to x i . Therefore, the interactions between the ith and jth cortical columns are possible in this case. We conjecture that they interact via phase deviations [1]. A direct consequence of this fact is the following result. Conclusion. Hoppensteadt and Izhikevich [1] The cortical columns can be divided into groups, or ensembles, according to their frequencies Omega i . Columns from different ensembles have different frequencies and interactions between ....

....synaptic connections from x j to x i . Therefore, the interactions between the ith and jth cortical columns are possible in this case. We conjecture that they interact via phase deviations [1] A direct consequence of this fact is the following result. Conclusion. Hoppensteadt and Izhikevich [1]) The cortical columns can be divided into groups, or ensembles, according to their frequencies Omega i . Columns from different ensembles have different frequencies and interactions between them are negligible (see Fig.3) Thus, no matter what kind of synaptic connections there are between ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model. Biological Cybernetics, 75:117--127, 1996

....fi i is still an open problem. 3. Cohen Grossberg Convergence Theorem The following result is a generalization of the celebrated Cohen Grossberg Convergence Theorem [4] for oscillatory neural networks, and it is relevant to the rotation vector method [5] Theorem. Hoppensteadt and Izhikevich [7]) Suppose all oscillators in the canonical model z 0 i = ae i i i )z i Gamma z i jz i j 2 n X j=1 c ij z j have equal frequencies 1 = Delta Delta Delta = i = and the matrix of complex valued synaptic connections is selfadjoint, i.e. c ij = c ji for all i and j, where ....

....cycle e i z . Obviously, any pair of oscillators has a constant (non necessarily zero) phase difference on this limit cycle. 2 4. Hebbian Learning Rule. The self adjoint matrix of synaptic coefficients can arise as a result of the Hebbian learning rule. The rule in this case has the form [7] c 0 ij = Gammafl c ij kz i z j ; where fl 0 is the fading constant and k 0 is a parameter. Notice that the complex conjugate activity of the jth oscillator, namely z j , is used here. This guarantees that the network can learn and correctly reproduce phase deviations, which look ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: II. Learning of Phase Information. Biological Cybernetics, 75:129--135, 1996

....can significantly distort the oscillations or even kill them [2] It should be stressed that each synaptic coefficient c ij encodes two attributes: The strength of connection s ij = jc ij j and the phase of connection ij = Arg c ij . The former has obvious meaning, the latter is scrutinized in [6]. Both numbers play essential role in memorization and recognition. The parameter fi i describes how the frequency depends on the amplitude. In what follows we take fi i = 0. Studying neuro computational properties of canonical model (2) for non zero fi i is still an open problem. 3. ....

F.C. Hoppensteadt and E.M. Izhikevich. Synaptic Organizations and Dynamical Properties of Weakly Connected Neural Oscillators: I. Analysis of Canonical Model. Biological Cybernetics, 75:117--127, 1996

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