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P3: Using Higher-Order Networks to Analyze Non-Linear Dynamics in Neural Network Models
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**Xerxes Arsiwalla**

In linear dynamical systems, one has an elegant way to analyze the system's dynamics using a network representation of the state transition matrix, obtained from a state space formulation of the system of ODEs. However, in non-linear systems, there is no state space formulation to begin with. In recent work, we have established a correspondence between non-linear dynamical systems and higher-order networks [1] (see also [2]). The latter refer to graphs that include links between nodes and edges, as well as links between two edges. It turns out that such networks have a rich structure capable of representing non-linearities in the vector field of dynamical systems. To do this, one has to first dimensionally unfold a system of non-linear ODEs such that non-linear terms in the vector field can be re-expressed using auxiliary dynamical variables. This results in an unfolded dynamical system with only polynomial non-linearities. This operation works for a large class of non- linear systems. It turns out that once we have a polynomial vector field, the system can then be expressed in generalized state space form. This is what ultimately admits a graphical representation of the system. However, the resulting graph consists of higher-order edges. This generalizes the more common usage of networks with dyadic edges to networks with compounded edges. Here, we show an application of these graphs to analyze neural networks built from sigmoidal rate models as well as mean-field models. Higher-order graphs enable one to systematically decompose contributions of various non-linear terms to the dynamics as well as analyze stability and control of the system using network properties.

References

1\. Arsiwalla, X.D., A Functorial Correspondence Between Non-Linear Dynamical Systems and Higher-Order Networks, Submitted (2020)

2\. Baez, J.C., Pollard, B.S., A compositional framework for reaction networks, Reviews in Mathematical Physics, 29(09), 1750028 (2017)

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In linear dynamical systems, one has an elegant way to analyze the system's dynamics using a network representation of the state transition matrix, obtained from a state space formulation of the system of ODEs. However, in non-linear systems, there is no state space formulation to begin with. In recent work, we have established a correspondence between non-linear dynamical systems and higher-order networks [1] (see also [2]). The latter refer to graphs that include links between nodes and edges, as well as links between two edges. It turns out that such networks have a rich structure capable of representing non-linearities in the vector field of dynamical systems. To do this, one has to first dimensionally unfold a system of non-linear ODEs such that non-linear terms in the vector field can be re-expressed using auxiliary dynamical variables. This results in an unfolded dynamical system with only polynomial non-linearities. This operation works for a large class of non- linear systems. It turns out that once we have a polynomial vector field, the system can then be expressed in generalized state space form. This is what ultimately admits a graphical representation of the system. However, the resulting graph consists of higher-order edges. This generalizes the more common usage of networks with dyadic edges to networks with compounded edges. Here, we show an application of these graphs to analyze neural networks built from sigmoidal rate models as well as mean-field models. Higher-order graphs enable one to systematically decompose contributions of various non-linear terms to the dynamics as well as analyze stability and control of the system using network properties.

References

1\. Arsiwalla, X.D., A Functorial Correspondence Between Non-Linear Dynamical Systems and Higher-Order Networks, Submitted (2020)

2\. Baez, J.C., Pollard, B.S., A compositional framework for reaction networks, Reviews in Mathematical Physics, 29(09), 1750028 (2017)

Monday July 20, 2020 9:00pm - 10:00pm CEST

Slot 14

Slot 14