Dynamics of Evolving Feed-Forward Neural Networks and Their Topological Invariants

The evolution of a simulated feed-forward neural network with recurrent excitatory connections and inhibitory forward connections is studied within the framework of algebraic topology. The dynamics includes pruning and strengthening of the excitatory connections. The invariants that we define are based on the connectivity structure of the underlying graph and its directed clique complex. The computation of this complex and of its Euler characteristic are related with the dynamical evolution of the network. As the network evolves dynamically, its network topology changes because of the pruning and strengthening of the onnections and algebraic topological invariants can be computed at different time steps providing a description of the process. We observe that the initial values of the topological invariant computed on the network before it evolves can predict the intensity of the activity.

[1]  R. Ho.  Algebraic Topology , 2022 .

[2]  E. Vaadia,et al.  Spatiotemporal structure of cortical activity: properties and behavioral relevance. , 1998, Journal of neurophysiology.

[3]  Alessandro E. P. Villa,et al.  Effect of stimulus-driven pruning on the detection of spatiotemporal patterns of activity in large neural networks , 2007, Biosyst..

[4]  Danielle S. Bassett,et al.  Two’s company, three (or more) is a simplex , 2016, Journal of Computational Neuroscience.

[5]  Moritz Deger,et al.  Reconstruction of recurrent synaptic connectivity of thousands of neurons from simulated spiking activity , 2015, Journal of Computational Neuroscience.

[6]  Alessandro E. P. Villa,et al.  The topology of the directed clique complex as a network invariant , 2015, SpringerPlus.

[7]  Professor Moshe Abeles,et al.  Local Cortical Circuits , 1982, Studies of Brain Function.

[8]  Michael W. Reimann,et al.  Topological analysis of the connectome of digital reconstructions of neural microcircuits , 2016, 1601.01580.

[9]  Idan Segev,et al.  On the Transmission of Rate Code in Long Feedforward Networks with Excitatory–Inhibitory Balance , 2003, The Journal of Neuroscience.

[10]  Moshe Abeles,et al.  Corticonics: Neural Circuits of Cerebral Cortex , 1991 .

[11]  Peter A. Appleby,et al.  Triphasic spike-timing-dependent plasticity organizes networks to produce robust sequences of neural activity , 2012, Front. Comput. Neurosci..

[12]  Gábor Csárdi,et al.  The igraph software package for complex network research , 2006 .

[13]  Moshe Abeles,et al.  On Embedding Synfire Chains in a Balanced Network , 2003, Neural Computation.

[14]  Ole Tange,et al.  GNU Parallel: The Command-Line Power Tool , 2011, login Usenix Mag..

[15]  G. Petri,et al.  Homological scaffolds of brain functional networks , 2014, Journal of The Royal Society Interface.

[16]  A. Aertsen,et al.  Spiking activity propagation in neuronal networks: reconciling different perspectives on neural coding , 2010, Nature Reviews Neuroscience.

[17]  M. Abeles.  Local Cortical Circuits: An Electrophysiological Study , 1982 .

[18]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[19]  I. Tetko,et al.  Spatiotemporal activity patterns of rat cortical neurons predict responses in a conditioned task. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Jochen Triesch,et al.  Robust development of synfire chains from multiple plasticity mechanisms , 2014, Front. Comput. Neurosci..

[21]  W J Freeman.  Neural networks and chaos. , 1994, Journal of theoretical biology.