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Page 61 Li et al. Intell Robot 2021;1(1):58-83 I http://dx.doi.org/10.20517/ir.2021.08

represent the conductances of the potassium, sodium, and passive channels, respectively. Inspired from
this membrane model for dynamic ion exchanges, Grossberg proposed a shunting model [12,20,21] . By setting

= 1 and substituting = + , = , = + , = − , = , and = in

Equation (1), a shunting equation is obtained as [22,23]

= − + ( − ) − ( + ) (2)

where istheneuralactivity (membranepotential) ofthe -th neuron; , , and arenonnegativeconstants
representing the passive decay rate, the upper and lower bounds of the neural activity, respectively; and and

are the excitatory and inhibitory inputs to the neuron, respectively. In the shunting model, and are not

essential factors because the neural activity is the relative values between the boundary lines. Only parameter
determines the model dynamics. However, can be chosen in a very wide range. Thus, the shunting model
is not very sensitive to the model parameters [13] .
Equation (2) shows that the increase of activity depends on the positive term ( − ) that relies on

both the excitatory input and the difference of neural activity to its upper bound ( − ). Therefore, the

increases of become slower as the value of is closing to the upper bound . If the value of equals to ,
the ( − ) term becomes zero, and positive term has no effect no matter how big the excitatory input is.

In the case that the value of is greater than , the ( − ) term becomes negative, then the positive term
becomes negative, the excitatory input will decrease the activity until it is not higher than . Therefore, is
theupper boundoftheneuralactivity . The same forthenegativeterm ( + ) , whichguarantees thatthe

neural activity is always greater than the lower bound − . Thus, the neural activity is bounded between
the [− , ] region under various inputs conditions. The shunting model has been studied to understand the
adaptive behaviors of individuals in dynamic and complex environments [12] . Many achievements have been
accomplished in the past decades, such as, machine vision, sensory motor control, and many other areas [21,22] .
In the field of robotics, the shunting model has been wildly used in path planning, tracking control, hunting,
cooperation of various autonomous robots [13,24–26] .

2.2. Model variants
If the excitatory and inhibitory inputs in Equation (2) are lumped together and the auto-gain control terms are
removed, then Equation (2) can be written into a simpler form

= − + (3)

where is the total inputs of the -th neuron. Then, Equation (3) is rewritten as:

∑

= − + + ( ) (4)

=1
where is the connection weight from the -th neuron to the -th neuron; () is an activation function;
represents the external input to the -th neuron; and is the total number of neurons in the neural network.
In most situations, the additive model is computationally simpler and can also generate the real-time collision-
free path for robots. However, the shunting model has two important advantages. Firstly, the shunting model
inEquation(2)hasexcitatoryandinhibitoryauto-gaincontrolterms, ( − ) and ( + ), respectively, which
give the shunting model the dynamic responsive ability to input signals. The shunting model is more sensitive
to the changes of inputs [13] . Nevertheless, the dynamics of the additive model may saturate in some situations.
Secondly, the shunting model is bounded between the upper bound and lower bound − , whereas the
additive model is bounded only by limiting the input signals. The additive models have been widely applied to
artificial vision, learning-based algorithms, and other research fields [21] . Owning to the simple computation
process, even the limitations of the additive model exist, the additive model has been also applied to replace
the shunting model in many situations [13,15] .