Quantum Hopfield Model

The Hopfield model in a transverse field is investigated in order to clarify how quantum fluctuations affect the macroscopic behavior of neural networks. Using the Trotter decomposition and the replica method, we find that the α (the ratio of the number of stored patterns to the system size)-∆ (the strength of the transverse field) phase diagram of this model in the ground state resembles the α-T phase diagram of the Hopfield model quantitatively, within the replica-symmetric and static approximations. This fact suggests that quantum fluctuations play quite similar roles to thermal fluctuations in neural networks as long as macroscopic properties are concerned. PACS numbers: 05.50.+q, 87.10.+e, 75.10.Jm, 75.10.Nr Typeset using REVTEX 1 Neural networks have been investigated very actively in the context of physics and engineering in terms of simple mathematical models inspired by anatomical and physiological facts about the brain [1,2]. In these models, the state of a neuron is often described by an Ising spin, corresponding to the firing or at-rest state. Neurons are connected with each other by long-range interactions, and if these interactions are chosen suitably, some fixed patterns of spins remain dynamically stable. Thus, the system works as an associative memory. In the time-evolution process of real neurons, randomness appears in signal transmission at a synapse: A pulse reaching the terminal bulb of an axon does not always result in the release of neuro-transmitters contained in vesicles. Usually, this randomness in signal transmission has been taken into account in models as thermal fluctuations, which leads to a statistical-mechanical formulation of neural networks (see Sec. 2.1.3. of Ref. [2]). That is, the “Hamiltonian” H is defined so as to give stable fixed patterns of the relevant network as global or local minima of the energy landscape. The “temperature” T is next introduced and the “partition function” is defined by Z ≡ Tre for a given sample of embedded patterns. The “free energy” is then obtained from the quenched average over the samples as F = −T 〈〈logZ〉〉. However, detailed considerations of the origin of randomness in signal transmission suggest that quantum effects may be an important driving force to cause uncertainty in the release of neuro-transmitters from vesicles into the synaptic cleft. For example, Stapp pointed out [3] that stochastic characters of signal transmission at synapses may be explained by quantum uncertainty in the positions of calcium ions during migration in the terminal bulb of an axon. Beck and Eccles argued [4] that quantum fluctuations can be of comparable order as thermal fluctuations in the hydrogen bridges within axon terminals which control the exocytosis of synaptic vesicles. These investigations strongly indicate the necessity to treat randomness in the signal transmission in terms of quantum mechanics. Under these motivations, we investigate a neural network with quantum fluctuations. In the conventional statistical-mechanical approach to neural networks [1,2], the parameter (temperature) T is introduced into a Hamiltonian system. This parameter does not 2 necessarily reflect directly the detailed stochastic properties of the original time-dependent neuron system. Nevertheless, the effect of the parameter T is still regarded as a prototype of thermal fluctuations. Similarly, the introduction of quantum fluctuations at the level of the Hamiltonian formulation of the problem is expected to be helpful in clarifying the roles of quantum effects in the original system. Admittedly, the model defined below is not a faithful reproduction of real processes in the brain. However, our purpose is not to explain the brain itself in detail [5]. We rather aim to clarify the statistical-mechanical roles of quantum fluctuations inlarge-scale networks at a phenomenological level. We believe that our model serves as a first step toward this goal. Another motivation to develop the following argument is that our method of investigation provides a typical framework to treat quantum spin systems with quenched randomness. Let us therefore consider the Hopfield model [6] in a transverse field, H = − ∑ i,j i6=j Jijσ z i σ z j −∆ ∑ i σ i ≡ H0 +H1 , (1) with the synaptic weight Jij defined by the Hebb rule, Jij = 1 N p