(1 + ∞)-dimensional attractor neural networks

Skantzos, N. S.; Coolen, Anthony C. C.

doi:10.1088/0305-4470/33/33/301

Cited by 21 publications

(69 citation statements)

References 17 publications

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“…For the special case of p = 2 the paramagnetic and ferromagnetic phase coexist between the first and second-order transition lines with as tri-critical point βJ = √ 3 ≈ 1.732, βJ 0 = − log(3)/4 ≈ −0.275. The latter results are in agreement with the results of [20] and with those of [21] in the case of one dimension. This analysis serves as a limiting case of our small-world model for increasingly larger values of the mean connectivity per site c. …”

Section: The 1 + ∞ Dimensional Modelsupporting

confidence: 92%

“…We remark that this equation reduces to the one presented in [20] for p = 2. To find the phase diagram of this system we perform a bifurcation analysis.…”

Section: The 1 + ∞ Dimensional Modelmentioning

confidence: 80%

“…To proceed with the evaluation of the traces in (20) we remark that in the thermodynamic limit N → ∞ only λ 0 , the largest eigenvalue of T [F ] will contribute. Defining…”

Section: B Transfer-matrix Analysismentioning

confidence: 99%

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Thermodynamics of spin systems on small-world hypergraphs

2006

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We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size we are able to solve the model analytically. In the more general case where the number of connections is finite we determine the static and dynamic ferromagneticparamagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.

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Section: The 1 + ∞ Dimensional Modelsupporting

confidence: 92%

“…We remark that this equation reduces to the one presented in [20] for p = 2. To find the phase diagram of this system we perform a bifurcation analysis.…”

Section: The 1 + ∞ Dimensional Modelmentioning

confidence: 80%

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Thermodynamics of spin systems on small-world hypergraphs

2006

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“…However, to make further progress, we need quantitative tools that are able to handle the complexity of the immune system's intricate signalling patterns. Fortunately, over the last decades a powerful arsenal of statistical mechanical techniques was developed in the disordered system community to deal with heterogeneous many-variable systems on complex topologies [15,16,17,18,19]. In the present paper we exploit these new techniques to model the multitasking capabilities of the (adaptive) immune network, where effector branches (B-cells) and coordinator branches (T-cells) interact via (eliciting and suppressive) signaling proteins called cytokines.…”

Section: Introductionmentioning

confidence: 99%

Immune networks: multitasking capabilities near saturation

Agliari

Annibale

Barra

et al. 2013

J. Phys. A: Math. Theor.

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120

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Abstract. Pattern-diluted associative networks were introduced recently as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T-and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with N T Tlymphocytes can manage a number N B = O(N δ T ) of B-lymphocytes simultaneously, with δ < 1. Here we study this model in the extensive load regime N B = αN T , with also a high degree of pattern dilution, in agreement with immunological findings. We use graph theory and statistical mechanical analysis based on replica methods to show that in the finite-connectivity regime, where each T-lymphocyte interacts with a finite number of B-lymphocytes as N T → ∞, the T-lymphocytes can coordinate effective immune responses to an extensive number of distinct antigen invasions in parallel. As α increases, the system eventually undergoes a second order transition to a phase with clonal cross-talk interference, where the system's performance degrades gracefully. Mathematically, the model is equivalent to a spin system on a finitely connected graph with many short loops, so one would expect the available analytical methods, which all assume locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results are supported by numerical simulations.

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“…The former does not affect pattern retrieval qualitatively [34][35][36][37][38][39], whereas the latter causes a switch from serial to parallel processing [30,31] (i.e. to simultaneous pattern recall).…”

Section: Introductionmentioning

confidence: 99%

Immune networks: multi-tasking capabilities at medium load

Agliari

Annibale

Barra

et al. 2013

J. Phys. A: Math. Theor.

Self Cite

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Associative network models featuring multi-tasking properties have been introduced recently and studied in the low-load regime, where the number P of simultaneously retrievable patterns scales with the number N of nodes as P ∼ log N. In addition to their relevance in artificial intelligence, these models are increasingly important in immunology, where stored patterns represent strategies to fight pathogens and nodes represent lymphocyte clones. They allow us to understand the crucial ability of the immune system to respond simultaneously to multiple distinct antigen invasions. Here we develop further the statistical mechanical analysis of such systems, by studying the mediumload regime, P ∼ N δ with δ ∈ (0, 1]. We derive three main results. First, we reveal the nontrivial architecture of these networks: they exhibit a high degree of modularity and clustering, which is linked to their retrieval abilities. Second, by solving the model we demonstrate for δ < 1 the existence of large regions in the phase diagram where the network can retrieve all stored patterns simultaneously. Finally, in the high-load regime δ = 1 we find that the system behaves as a spin-glass, suggesting that finite-connectivity frameworks are required to achieve effective retrieval.

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(1 + ∞)-dimensional attractor neural networks

Cited by 21 publications

References 17 publications

Thermodynamics of spin systems on small-world hypergraphs

Thermodynamics of spin systems on small-world hypergraphs

Immune networks: multitasking capabilities near saturation

Immune networks: multi-tasking capabilities at medium load

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