# Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions

Wersing H, Beyn W-J, Ritter H (2001) *Neural Computation* 13(8): 1811-1825.

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We establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions, also called linear threshold or semilinear transfer functions. As Hahnloser, Sarpeshkar, Mahowald, Douglas, and Seung (2000) showed, networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit. To obtain this behavior, the network must be multistable and nondivergent, and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification. The first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron. The second condition is restricted to symmetric networks, but can also take into account the stabilizing effect of nonlocal inhibitory interactions. We demonstrate the application of the conditions on a simple example and the orientation-selectivity mo del of Ben-Yishai, Lev Bar-Or, and Sompolinsky (1995). We show that the conditions can be used to identify in their model regions of maximal orientation-selective amplification and symmetry breaking.

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Wersing H, Beyn W-J, Ritter H. Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions.

*Neural Computation*. 2001;13(8):1811-1825.Wersing, H., Beyn, W. - J., & Ritter, H. (2001). Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions.

*Neural Computation*,*13*(8), 1811-1825. doi:10.1162/08997660152469350Wersing, H., Beyn, W. - J., and Ritter, H. (2001). Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions.

*Neural Computation*13, 1811-1825.Wersing, H., Beyn, W.-J., & Ritter, H., 2001. Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions.

*Neural Computation*, 13(8), p 1811-1825. H. Wersing, W.-J. Beyn, and H. Ritter, “Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions”,

*Neural Computation*, vol. 13, 2001, pp. 1811-1825. Wersing, H., Beyn, W.-J., Ritter, H.: Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Computation. 13, 1811-1825 (2001).

Wersing, Heiko, Beyn, Wolf-Jürgen, and Ritter, Helge. “Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions”.

*Neural Computation*13.8 (2001): 1811-1825.
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### 26 Citations in Europe PMC

Data provided by Europe PubMed Central.

Attentional recruitment of inter-areal recurrent networks for selective gain control.

Hahnloser RH, Douglas RJ, Hepp K.,

PMID: 12079551

Hahnloser RH, Douglas RJ, Hepp K.,

*Neural Comput*14(7), 2002PMID: 12079551

### 18 References

Data provided by Europe PubMed Central.

On the piecewise analysis of networks of linear threshold neurons.

Hahnloser RL.,

PMID: 12662807

Hahnloser RL.,

*Neural Netw*11(4), 1998PMID: 12662807

A model for the depth-dependence of receptive field size and contrast sensitivity of cells in layer 4C of macaque striate cortex.

Bauer U, Scholz M, Levitt JB, Obermayer K, Lund JS.,

PMID: 10341989

Bauer U, Scholz M, Levitt JB, Obermayer K, Lund JS.,

*Vision Res.*39(3), 1999PMID: 10341989

A model for the intracortical origin of orientation preference and tuning in macaque striate cortex.

Adorjan P, Levitt JB, Lund JS, Obermayer K.,

PMID: 10367965

Adorjan P, Levitt JB, Lund JS, Obermayer K.,

*Vis. Neurosci.*16(2), 1999PMID: 10367965

Recurrent excitation in neocortical circuits.

Douglas RJ, Koch C, Mahowald M, Martin KA, Suarez HH.,

PMID: 7638624

Douglas RJ, Koch C, Mahowald M, Martin KA, Suarez HH.,

*Science*269(5226), 1995PMID: 7638624

Piecewise affine bijections of ?n, and the equation Sx+−Tx−=y

Kuhn,

Kuhn,

*Linear Algebra and its Applications*96(1), 1987
A competitive-layer model for feature binding and sensory segmentation.

Wersing H, Steil JJ, Ritter H.,

PMID: 11177439

Wersing H, Steil JJ, Ritter H.,

*Neural Comput*13(2), 2001PMID: 11177439

On Neurodynamics with Limiter Function and Linsker's Developmental Model

Feng,

Feng,

*Neural Computation*8(5), 1996Lippmann,

*IEEE ASSP Magazine*4(2), 1987

Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit.

Hahnloser RH, Sarpeshkar R, Mahowald MA, Douglas RJ, Seung HS.,

PMID: 10879535

Hahnloser RH, Sarpeshkar R, Mahowald MA, Douglas RJ, Seung HS.,

*Nature*405(6789), 2000PMID: 10879535

A model of multiplicative neural responses in parietal cortex.

Salinas E, Abbott LF.,

PMID: 8876244

Salinas E, Abbott LF.,

*Proc. Natl. Acad. Sci. U.S.A.*93(21), 1996PMID: 8876244

Computational differences between asymmetrical and symmetrical networks.

Li Z, Dayan P.,

PMID: 10372762

Li Z, Dayan P.,

*Network*10(1), 1999PMID: 10372762

Feng,

*Journal of Physics A Mathematical and General*29(16), 1996

Convergent activation dynamics in continuous time networks

Hirsch,

Hirsch,

*Neural Networks*2(5), 1989
Equilibria of the brain-state-in-a-box (BSB) neural model

Greenberg,

Greenberg,

*Neural Networks*1(4), 1988
Spatial summation of inhibitory influences in the eye of Limulus, and the mutual interaction of receptor units.

HARTLINE HK, RATLIFF F.,

PMID: 13525682

HARTLINE HK, RATLIFF F.,

*J. Gen. Physiol.*41(5), 1958PMID: 13525682

Self-organization of orientation sensitive cells in the striate cortex.

von der Malsburg C.,

PMID: 4786750

von der Malsburg C.,

*Kybernetik*14(2), 1973PMID: 4786750

Lyapunov functions for neural nets with nondifferentiable input-output characteristics.

Feng J.,

PMID: 9117900

Feng J.,

*Neural Comput*9(1), 1997PMID: 9117900

Theory of orientation tuning in visual cortex.

Ben-Yishai R, Bar-Or RL, Sompolinsky H.,

PMID: 7731993

Ben-Yishai R, Bar-Or RL, Sompolinsky H.,

*Proc. Natl. Acad. Sci. U.S.A.*92(9), 1995PMID: 7731993

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