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Quantitative study
Eect of dimension Concerning the replica calculation, the two-dimensional case does not dier much from the one-dimensional one. It is detailed in [1], Appendix A. Quali- tatively, the phase diagram is not expected to dier, though we have not computed it in details, except the clump phase stability region. Quantitatively, for equal f and w, the clump phase is slightly reduced in 2D as compared to 1D (see Figure 23 in [1]). We expect the three phases to exist in any dimension of space, although the calculation would be more dicult.
Eect of parameters We have also looked at the in uence of the model’s parameters f and w on the phase diagram, especially on the clump phase domain, which is the phase of interest. For this we consider critical quantities such as TCL, CL, Tc and g (see Figure 2.2’s caption). We compute these quantities for varying f or w. The result is shown in Figures 18, 19, 20 and 21 in [1], Appendix A. Regarding the critical temperatures Tc and TCL, the dependence on f and w is monotonic. A more interesting eect appears with the critical loads g and CL which exhibit a maximum for values around roughly f w. This optimum can be understood as a tradeo between sparsity and limiting the cross-talk with other stored environments.
Silent cells Finally, we have extended our calculation to take into account \silent cells » (Section 1.4.2). We incorporate in the averaging procedure the random selection of a given fraction c of cells having a place eld in a given environment (0 < c < 1), the others cells being silent. The calculation, detailed in [1], Appendix A, leads to a simple modication of the free-energy functional. It turns out that this additional hypothesis alters the phase diagram only quantitatively, with Tc and g increasing monotonically with c (see Fig. 22 in [1]).
Parallel with the Hopeld model phase diagram
It is instructive to compare the phase diagram we have obtained (Fig. 2.2) with the Hopeld model’s [10, 90] (Fig. 2.4).
Common points The Hopeld model also exhibit three phases | paramagnetic, spin glass and ferromagnetic (‘retrieval’, where the stored patterns are attractors) | with an obvious parallel with our paramagnetic, spin glass and clump phases respectively. The retrieval phase is also replica-symmetry broken in a low-T, high- region, though this region is much smaller than in the present case. As in our model, there is a second order transition between the paramagnetic and spin glass phases, along a line scaling as p. Another common point is that in both models the spin glass phase is always unstable against replicon modes.
As regards the derivation of this phase diagram, the replica calculation performed by Amit, Gutfreund and Sompolinsky [90] follows basically the same steps as ours | aver- age over disorder of the partition function replicated n times, introduction of conjugated parameters, saddle-point equations, replica-symmetric Ansatz. Our isolating of the ‘ref- erence environment’ by the indexing choice is replaced, in the case of discrete patterns, by an external symmetry-breaking eld that selects the pattern(s) to retrieve and that is nally sent to zero.
Dierences The main dierence concerns the shape of the ferromagnetic/clump do- main. In the Hopeld model, it does not overlap with the paramagnetic domain, so these two phases do not coexist. Moreover, in the Hopeld model it has a triangular shape (the storage capacity decreases monotonically with the temperature) so there is no reentrance as in our case. A very weak reentrance was nevertheless found by subsequent studies [91], but in a much lesser extent. This can be interpreted in terms of robustness to noise. Intuitively, noise can much more easily disrupt a point attractor than a continuous one. Our phase diagram shows that a moderate level of noise can even increase the storage capacity.
Emergence of a ‘quasi-particle’: demonstration in the = 0 case
By translational invariance of the saddle-point equations, in the clump phase the bump of activity can concentrate equivalently around any position of the environment. It is then intuitive to consider the bump as a solid and to describe its motion in this map with a single coordinate located at its centre. In other words, it seems natural to extend the macroscopic description of the activity established at equilibrium to the dynamics.
This kind of description has already been adopted before (see for instance [82]). Yet, it was then postulated without demonstration: the fact that noise makes the clump move globally with little deformation had not been proven so far.
In the case of our model without disorder (single-environment case, = 0) and in the large N limit, we have analytically derived this quasi-particle behaviour6. More precisely, we have shown that the microscopic stochastic evolutions of the units result in a collective pure diusion motion of the clump; while the shape of the clump only shows small uc- tuations around its equilibrium. The position of the centre of the clump, xc(t) (collective coordinate), obeys the diusion equation x_c(t) = p D0(t) .
Competition between transition and diusion
This point is a very robust feature of the model: an increase in the diusion constant is always accompanied by an increase in the rate of transitions to other environments. We have tested several ways to improve the motility of the clump: besides moving to the edge of the clump’s stability domain (by increasing or T or by lowering c), we have tried additional, out-of-equilibrium mechanisms such as adaptation, theta rhythm or asymmetric dilution of the synapses. They are described in [2]. We observe that, whatever the mechanism, as soon as diusion is enhanced, so are transitions.
Two simple explanations can be proposed for that. First, facilitating diusion basically means, one way or another, increasing the noise level. So the clump is destabilized and more likely to transit to another map. Then, the more the bump diuses, the more it is susceptible to nd a favourable position for transition (see Section 2.4). These two scenarii are not mutually exclusive, and it seems reasonable to say that they both contribute to the ubiquitous competition between the two dynamics.
Table of contents :
Introduction
1 Computational Neuroscience, Statistical Mechanics and the hippocam- pus
1.1 Methods in Computational Neuroscience
1.2 The Statistical Mechanics Approach
1.3 The Attractor Neural Network Theory
1.4 The Hippocampus
2 A model for place cells
2.1 On models
2.2 Description
2.3 Phase diagram
2.4 Transitions between maps
2.5 Diusion within one map
2.6 Comparison with previous models
2.7 Estimation of the parameters from experimental data
3 Decoding
3.1 Stating the decoding problem
3.2 Experimental data
3.3 Rate-based decoding methods
3.4 The inverse-Ising approach to decoding
3.5 Structure of the inferred eective networks
3.6 Decoding an environment
3.7 Decoded transitions between maps
3.8 Correspondence between the \Ising – coupled » and the \rate – max. poste- rior »methods
4 Conclusions & Extensions
4.1 Summary of results
4.2 Rening the model
4.3 Future work on the decoding issue
4.4 Concluding remarks
Appendix A Publications
Appendix B Resume en francais
Bibliography
