A simple model of a neuron with self-interaction is described by Pasemann

[12]. The difference equation is given by an+1 = γan + θ + wσ(an), 0 ≤ γ < 1, where an is the activation level of the neuron, θ is a bias, w is a self-weight,

γ represents dissipation in a neuron, and the output is given by the sigmoidal transfer function

σ(x) =

1 1 + e−x

.

(a) Determine an equation for the fixed points of period one and show that the stability condition is given by |γ +wσ′(a)| < 1, where a is a fixed point of period one.

(b) Show that the system is bistable in the region bounded by the parametric equations

θ

(a) = (1 − γ )a −

(1 − γ )

(1 − σ(a))

, w

(a) =

(1 − γ )

σ′(a)

.

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(c) Show that the system is unstable in the region bounded by the parametric equations θ

(a) = (1 − γ )a +
(1 + γ )
(1 − σ(a))
, w

(a) = −
(1 + γ )
σ′(a)
.

(d) Use the first iterative method to plot a bifurcation diagram when θ = 4 and w = −16 for 0 < γ < 1.

(e) Use the second iterative method to plot a bifurcation diagram when θ = −2.4 and γ = 0 for 3 < w < 7. Ramp w up and down.