MACHINE LEARNING TECHNIQUES - LASA

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The continuous time Hopfield network is a powerful to tool to store sequences. It has been used
in numerous applications in robotics, e.g. learning to recognize and reproduce complex patterns
of motion. The strength of this modeling comes from the fact that the complexity of the pattern
that emerges from mixing different first order differential equations as given in (6.76) becomes
very rapidly untractable.
Consider the case whereby the neuron received a single steady input S with no self-connection,
i.e. with no decay term. The behavior of such a neuron is a linear differential equation that can be
solved analytically and is given by:
−t
/ τ
mt () = ( m0
− S)
e + S
1
(6.78)
xt () =
−D⋅ ( m( t) + b)
1 + e
The system grows until reaching a maximum as shown in Figure 6-16.
Figure 6-16: Dynamics of a single Leaky-Integrator neuron with no self-connection
The behavior of a single leaky-integrator neuron with a self-connection is much more complex. It
follows a nonlinear differential equation that cannot be solved analytically. To study convergence
of the neuron, one must then rely on finding the equilibrium points of the activation function of its
membrane potential. This is given by:
dm
= 0 ⇒ m− w11σ
( m+ b)
= S
dt
(6.79)
1
where σ ( z) = is the sigmoid function
−Dz
1 + e
© A.G.Billard 2004 – Last Update March 2011