Team:USP-UNESP-Brazil/Associative Memory/Modeling

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A third equilibrium point emerges when $\frac{\phi_A}{\phi_B} \gg 1$ - which means the repression of population A over population B is much greater than its counterpart, Fig. 3. In this case, the system reaches an equilibrium close to $(1,0)$: population A activated, population B repressed. The behavior is analogous if $\frac{\phi_A}{\phi_B} \ll 1$.
A third equilibrium point emerges when $\frac{\phi_A}{\phi_B} \gg 1$ - which means the repression of population A over population B is much greater than its counterpart, Fig. 3. In this case, the system reaches an equilibrium close to $(1,0)$: population A activated, population B repressed. The behavior is analogous if $\frac{\phi_A}{\phi_B} \ll 1$.
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<h1 id="discussion">Discussion</h1>
<h1 id="discussion">Discussion</h1>
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The Associative Memory Network project was based on a mathematical formulation of a neural network developed in 1982 by John Hopfield [3]. In order to connect the bacteria behavior during quorum sensing to a Hopfield network, we introduced an interaction between two populations in a mathematical model for quorum sensing, in order to represent neuronal repression/activation. Through steady-state analysis, it was possible to find up to four equilibrium points, representing the activation of both populations, activation of one population and repression of the other, and repression of both populations. The existence of these steady-state solutions depends on the set of parameters, and stability analysis is being conducted to answer which regions in parameter space guarantee stability of each equilibria.
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The Associative Memory Network project was based on a mathematical formulation of a neural network developed in 1982 by John Hopfield [3]. In order to connect the bacteria behavior during quorum sensing to a Hopfield network, we introduced an interaction between two populations in a mathematical model for quorum sensing. This interaction represents the neuronal repression/activation. Through steady-state analysis, it was possible to find up to four equilibrium points, representing the activation of both populations, activation of one population and repression of the other, and repression of both populations. The existence of these steady-state solutions depends on the set of parameters and stability analysis is being conducted to answer which regions in parameter space guarantee stability of each equilibria.
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However, numerical simulations indicate that the carrying capacity $K$ is a fundamental parameter in the determination of the stability of quorum-state fixed points: changing its value had no effect on the location of these equilibria in $(x,y)$ space, but Figure 1 shows increasing $K$ leads to quorum. Therefore, even though $K$ does not affect the existence of any fixed points, their stability seems to depend on its value - it would change a steady state from unstable to stable, so that for a population that would not reach quorum at a low carrying capacity might reach it when this parameter is increased.
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However, numerical simulations indicate that the carrying capacity $K$ is a fundamental parameter in the determination of the stability of quorum-state fixed points: changing its value had no effect on the location of these equilibria in $(x,y)$ space but, as shown in figure 1, increasing $K$ leads to quorum. Therefore, even though $K$ does not affect the existence of any fixed points, their stability seems to depend on its value - it would change a steady state from unstable to stable, so that for a population that would not reach quorum at a low carrying capacity might reach it when this parameter is increased.
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In summary, the goal has been achieved: it was proven that two bacterial populations are able to interact through repression and activation in order to reproduce a Hopfield Network, and that different combinations of parameters and initial conditions may lead to different activation patterns: both populations activated, both repressed, or the asymetrical case - one up, one down.
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In summary, the goal of this mathematical modeling has been achieved: it was verified that two bacterial populations are able to interact through repression and activation in order to reproduce a Hopfield Network. Additionally, different combinations of parameters and initial conditions may lead to different activation patterns: both populations activated, both repressed, or the asymetrical case - one up, one down.
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Our further steps are to find the conditions for stability, and to investigate patterns on the initial conditions and the stable steady states - in other words, input and output. A network with more than two interacting populations holds a systemic memory capable of storing and responding to a much wider range of patterns - a possibility that is being now explored by our team.
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Our further steps are to find the conditions for stability which means to investigate the behaviour of the stable steady states as a function of the initial conditions - in other words, how the input changes the output. A network with more than two interacting populations is able to hold a systemic memory capable of storing and responding to a much wider range of patterns - a possibility that should be explored by our team in the future.

Revision as of 00:29, 27 September 2012