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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] .
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