. Proof, It follows that f ? (T ) = g(T ) ? f (T ) = f (T ) ? g(T ) since f T (T ) = 0 by definition of the minimal polynomial. Also, since f ? divides f , there is k ? R[x] so that f = kf ? and thus f, For the statement about images we have f (T ) = f ? (T ) ? k(T ) =? ?(f (T )) ? ?(f ? (T ))

. Lemma-8, Let v ? R n , then f v = f

. Proof, The first statement follows as if f (T ) annihilates v then it also annihilates g(T )(v) for any g ? R[x]. The second statement follows since, by the first isomorphism theorem, we have

. Lemma-9, Let v ? R n and f, g ? R[x] with f g = f v . Then f w = f for w = g(T )(v)

]. P. Ashwin, O. Karabacak, and T. Nowotny, Criteria for robustness of heteroclinic cycles in neural microcircuits, The Journal of Mathematical Neuroscience, vol.1, issue.1, p.13, 2011.
DOI : 10.1186/2190-8567-1-13

C. Bick, C. , and M. I. Rabinovich, Dynamical Origin of the Effective Storage Capacity in the Brain???s Working Memory, Physical Review Letters, vol.103, issue.21, p.218101, 2009.
DOI : 10.1103/PhysRevLett.103.218101

P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcation and Dynamical Systems, Advanced Series in Nonlinear Dynamics 15, World Scientific, 2000.

C. Zhang, G. Dangelmayr, and I. Oprea, Storing cycles in Hopfield-type networks with pseudoinverse learning rule: Admissibility and network topology, Neural Networks, vol.46, pp.283-298, 2013.
DOI : 10.1016/j.neunet.2013.06.008

C. Zhang, G. Dangelmayr, and I. Oprea, Storing cycles in Hopfield-type networks with pseudoinverse learning rule: retrievability and bifurcation analysis, 2013.

S. David, R. M. Dummit, and . Foote, Abstract Algebra 3rd Edition, 2003.

T. Fukai and S. Tanaka, A Simple Neural Network Exhibiting Selective Activation of Neuronal Ensembles: From Winner-Take-All to Winners-Share-All, Neural Computation, vol.43, issue.1, pp.77-97, 1997.
DOI : 10.1162/neco.1989.1.3.334

T. Gencic, M. Lappe, G. Dangelmayr, and W. Guettinger, Storing cycles in analog neural networks. Parallel processing in neural systems and computers, pp.445-450, 1990.

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, 1998.
DOI : 10.1017/CBO9781139173179

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, pp.2554-2558, 1982.

M. Krupa, Robust heteroclinic cycles, Journal of Nonlinear Science, vol.2, issue.4, pp.129-176, 1997.
DOI : 10.1007/BF02677976

L. Personnaz, I. Guyon, and &. G. Dreyfus, Collective computational properties of neural networks: New learning mechanisms, Physical Review A, vol.34, issue.5, pp.4217-4228, 1986.
DOI : 10.1103/PhysRevA.34.4217

M. I. Rabinovich, V. S. Afraimovich, C. Bick, and P. Varona, Information flow dynamics in the brain, Physics of Life Reviews, vol.9, issue.1, pp.51-73, 2012.
DOI : 10.1016/j.plrev.2011.11.002

A. Szucs, R. Huerta, M. I. Rabinovich, and A. I. Selverston, Robust Microcircuit Synchronization by Inhibitory Connections, Neuron, vol.61, issue.3, pp.439-453, 2009.
DOI : 10.1016/j.neuron.2008.12.032