7,136 research outputs found
Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology
Cyclic patterns of neuronal activity are ubiquitous in animal nervous
systems, and partially responsible for generating and controlling rhythmic
movements such as locomotion, respiration, swallowing and so on. Clarifying the
role of the network connectivities for generating cyclic patterns is
fundamental for understanding the generation of rhythmic movements. In this
paper, the storage of binary cycles in neural networks is investigated. We call
a cycle admissible if a connectivity matrix satisfying the cycle's
transition conditions exists, and construct it using the pseudoinverse learning
rule. Our main focus is on the structural features of admissible cycles and
corresponding network topology. We show that is admissible if and only
if its discrete Fourier transform contains exactly nonzero
columns. Based on the decomposition of the rows of into loops, where a
loop is the set of all cyclic permutations of a row, cycles are classified as
simple cycles, separable or inseparable composite cycles. Simple cycles contain
rows from one loop only, and the network topology is a feedforward chain with
feedback to one neuron if the loop-vectors in are cyclic permutations
of each other. Composite cycles contain rows from at least two disjoint loops,
and the neurons corresponding to the rows in from the same loop are
identified with a cluster. Networks constructed from separable composite cycles
decompose into completely isolated clusters. For inseparable composite cycles
at least two clusters are connected, and the cluster-connectivity is related to
the intersections of the spaces spanned by the loop-vectors of the clusters.
Simulations showing successfully retrieved cycles in continuous-time
Hopfield-type networks and in networks of spiking neurons are presented.Comment: 48 pages, 3 figure
Schrieffer-Wolff transformation for quantum many-body systems
The Schrieffer-Wolff (SW) method is a version of degenerate perturbation
theory in which the low-energy effective Hamiltonian H_{eff} is obtained from
the exact Hamiltonian by a unitary transformation decoupling the low-energy and
high-energy subspaces. We give a self-contained summary of the SW method with a
focus on rigorous results. We begin with an exact definition of the SW
transformation in terms of the so-called direct rotation between linear
subspaces. From this we obtain elementary proofs of several important
properties of H_{eff} such as the linked cluster theorem. We then study the
perturbative version of the SW transformation obtained from a Taylor series
representation of the direct rotation. Our perturbative approach provides a
systematic diagram technique for computing high-order corrections to H_{eff}.
We then specialize the SW method to quantum spin lattices with short-range
interactions. We establish unitary equivalence between effective low-energy
Hamiltonians obtained using two different versions of the SW method studied in
the literature. Finally, we derive an upper bound on the precision up to which
the ground state energy of the n-th order effective Hamiltonian approximates
the exact ground state energy.Comment: 47 pages, 3 figure
- …