A Dynamical System Approach to
Modeling Mental Exploration
Giorgio Colangelo
Prof. Andreas Herz, Prof. Leone Fronzoni, Dr. Martin Stemmler
The hippocampal-entorhinal complex plays an essential role within the brain in spatial navigation, mapping a spatial path onto a sequence of cells that fire action potentials. During rest or sleep, these sequences are replayed in either re-verse or forward temporal order [1, 2]; in some cases, novel sequences occur that may represent paths not yet taken, but connecting contiguous spatial locations. These sequences potentially play a role in the planning of future paths. In par-ticular, mental exploration is needed to discover short-cuts or plan alternative routes.
Hopfield proposed a two-dimensional planar attractor network as a substrate for the mental exploration [3]. He extended the concept of a line attractor used for the ocular-motor apparatus [4], to a planar attractor that can memorize any spatial path and then recall this path in memory. Such a planar attractor contains an infinite number of fixed points for the dynamics, each fixed point corresponding to a spatial location.
For symmetric connections in the network, the dynamics generally admits a Lyapunov energy function L. Movement through different fixed points is possible because of the continuous attractor structure. In this model, a key role is played by the evolution of a localized activation of the network, a “bump”, that moves across this neural sheet that topographically represents space. For this to occur, the history of paths already taken is imprinted on the synaptic couplings between the neurons. Yet attractor dynamics would seem to preclude the bump from moving; hence, a mechanism that destabilizes the bump is required. The mechanism to destabilize such an activity bump and move it to other locations of the network involves an adaptation current that provides a form of delayed inhibition.
Both a spin-glass and a graded-response approach are applied to investigat-ing the dynamics of mental exploration mathematically. Simplifyinvestigat-ing the neural network proposed by Hopfield to a spin glass [5, 6], I study the problem of re-calling temporal sequences and explore an alternative proposal, that relies on storing the correlation of network activity across time, adding a sequence tran-sition term to the classical instantaneous correlation term during the learning of the synaptic couplings. In this approach, the biological “adaptation current” is interpreted as a local field that can destabilize the equilibrium causing the bump to move. We can also combine the adaptation and transition term to show how the dynamics of exploration is affected. To obtain goal-directed searching, we introduce a weak external field associated with a rewarded location. We show how the bump trajectory then follows a suitable path to get to the target.
For networks of graded-response neurons with weak external stimulation, amplitude equations known from pattern formation studies [7] in bio-chemico-physical systems are developed. This allows me to predict the modes of net-work activity that can be selected by an external stimulus and how these modes evolve. Using perturbation theory and coarse graining [8], the dynamical equa-tions for the evolution of the system are reduced from many sets of nonlinear integro-differential equations for each neuron to a single macroscopic equation. This equation, in particular close to the transition to pattern formation, takes the form of the Landau Ginzburg equation.
The parameters for the connections between the neurons are shown to be related to the parameters of the Landau-Ginzburg equation that governs the bump of activity. The role of adaptation within this approximation is studied, which leads to the discovery that the macroscopic dynamical equation for the system has the same structure of the coupled equations used to describe the propagation of the electrical activity within one single neuron as given by the Fitzhugh-Nagumo equations [9].
References
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[2] Thomas J Davidson, Fabian Kloosterman, and Matthew A Wilson. Article hippocampal replay of extended experience. Neuron, 63(4):497–507, 2009. [3] Hopfield J. J. Neurodynamics of mental exploration. PNAS, 107(4):1648–
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[5] Hopfield J. J. Neural networks and physical systems with emergent collective computational abilities. PNAS, 79(8):2554–2558, April 1982.
[6] Hopfield J. J. Neurons with graded response have collective computational properties like those of two-state neurons. PNAS, 81(10):3088–3092, 1984. [7] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium.
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[8] Bressloff P. C. Pattern formation in visual cortex. Les Houches Lectures in Neurophysics, 2005.
[9] R. FitzHugh. Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol., 17(4):257–278, 1955.
