The excerpt note is from the Michael’s book. The example code is implemented by Fangwen, which can help us to understand the process and principle of the method.

Michael Milford. Robot Navigation from Nature: Simultaneous Localisation, Mapping, and Path Planning Based on Hippocampal Models. Springer-Verlag Berlin Heidelberg Press, pp. 68-69, 2008.

To achieve stable activity profiles in the head direction network, a system of localised excitation and global inhibition is used. The internal dynamics of the head direction cells are shown in Fig. 7.8.

Fig. 7.8. Head direction cells excite themselves and other cells. The total activity sum is used to inhibit all cells equally.

Localised Excitation

Each cell is connected to all the cells in the network including itself via excitatory links. The self-connected weight is the largest, with weight values dropping off in a discrete Gaussian manner for connections to more distant cells. The matrix storing the excitatory weight values between all cells, , is given by:

where is the variance of the Gaussian distribution used to form the weights profile.

During an experiment, activity in each of the head direction cells is projected through the appropriate excitatory links back into the head direction cells. The updated cell activity level after excitation, , is given by:

where is the excitatory cell and is the excitatory link value from cell to cell .

Global Inhibition

Global inhibition based on the total activity in the network is performed to ensure that without input from allothetic stimuli, smaller activity packets will gradually ‘die out’. The updated activity level of a cell after inhibition, , is given by:

Global Normalisation

To ensure that the total activity sum in the head direction network stays constant, a global normalisation step is performed after the allothetic association, allothetic calibration, and ideothetic update steps. The cell activity levels after normalisation, ,are given by:

clear all;

global HDCELLS;                                % The hdcells network
HDCELLS = zeros(1,36);

% HDCELLS(15)=1;
HDCELLS(11)=0.05;
HDCELLS(12)=0.20;
HDCELLS(13)=0.25;
HDCELLS(14)=0.80;
HDCELLS(15)=0.28;
HDCELLS(16)=0.22;
HDCELLS(17)=0.16;
HDCELLS(18)=0.08;

global HD_DIMENSION_THETA;                     % The hdcell theta dimension
HD_DIMENSION_THETA = 36;

theta_size = 2*pi/HD_DIMENSION_THETA;  % 0.6283 radian  36 deg

global HD_TH_SUM_SIN_LOOKUP;
global HD_TH_SUM_COS_LOOKUP;
HD_TH_SUM_SIN_LOOKUP = sin((1:HD_DIMENSION_THETA).*theta_size);
HD_TH_SUM_COS_LOOKUP = cos((1:HD_DIMENSION_THETA).*theta_size);

global HD_CELLS_TO_AVG;
global HD_AVG_TH_WRAP;
HD_CELLS_TO_AVG = 3;
HD_AVG_TH_WRAP = [(HD_DIMENSION_THETA - HD_CELLS_TO_AVG + 1):HD_DIMENSION_THETA 1:HD_DIMENSION_THETA 1:HD_CELLS_TO_AVG];

global HD_WEIGHT_EXCITATION_DIMENSION;
HD_WEIGHT_EXCITATION_DIMENSION = 7;

global HD_WEIGHT_INHIBITION_DIMENSION;
HD_WEIGHT_INHIBITION_DIMENSION = 5;

global HD_WEIGHT_EXCITATION_DIMENSION_HALF;
global HD_WEIGHT_INHIBITION_DIMENSION_HALF;
HD_WEIGHT_EXCITATION_DIMENSION_HALF = floor(HD_WEIGHT_EXCITATION_DIMENSION/2);
HD_WEIGHT_INHIBITION_DIMENSION_HALF = floor(HD_WEIGHT_INHIBITION_DIMENSION/2);

global HD_WEIGHT_EXCITATION_VARIANCE;
HD_WEIGHT_EXCITATION_VARIANCE = 2;

global HD_WEIGHT_INHIBITION_VARIANCE;
HD_WEIGHT_INHIBITION_VARIANCE =3;

global HD_GLOBAL_INHIBITION;
HD_GLOBAL_INHIBITION = 0.00002;

global HD_WEIGHT_EXCITATION;                   % the hdcell local excitation weight matricies
global HD_WEIGHT_INHIBITION;                   % the hdcell local inhibition weight matricies
HD_WEIGHT_EXCITATION = test_create_hdcell_weights(HD_WEIGHT_EXCITATION_DIMENSION,HD_WEIGHT_EXCITATION_VARIANCE);
HD_WEIGHT_INHIBITION = test_create_hdcell_weights(HD_WEIGHT_INHIBITION_DIMENSION,HD_WEIGHT_INHIBITION_VARIANCE);

%% Local Excitation

global HD_E_TH_WRAP;
HD_E_TH_WRAP = [(HD_DIMENSION_THETA - HD_WEIGHT_EXCITATION_DIMENSION_HALF + 1):HD_DIMENSION_THETA 1:HD_DIMENSION_THETA 1:HD_WEIGHT_EXCITATION_DIMENSION_HALF];

hdcells_le_new = zeros(1,HD_DIMENSION_THETA);
for z=1:HD_DIMENSION_THETA
	if HDCELLS(z) ~= 0
       	hdcells_le_new(HD_E_TH_WRAP(z:z+HD_WEIGHT_EXCITATION_DIMENSION-1)) = ...
        	hdcells_le_new(HD_E_TH_WRAP(z:z+HD_WEIGHT_EXCITATION_DIMENSION-1))+ HDCELLS(z).*HD_WEIGHT_EXCITATION ;
    end
end
HDCELLS = HDCELLS + hdcells_le_new;


global HD_I_TH_WRAP;
HD_I_TH_WRAP = [(HD_DIMENSION_THETA - HD_WEIGHT_INHIBITION_DIMENSION_HALF + 1):HD_DIMENSION_THETA 1:HD_DIMENSION_THETA 1:HD_WEIGHT_INHIBITION_DIMENSION_HALF];


%% local inhibition 
hdcells_li_new = zeros(1,HD_DIMENSION_THETA);
for z=1:HD_DIMENSION_THETA
   if HDCELLS(z) ~= 0
       hdcells_li_new(HD_I_TH_WRAP(z:z+HD_WEIGHT_INHIBITION_DIMENSION-1)) = ...
       hdcells_li_new(HD_I_TH_WRAP(z:z+HD_WEIGHT_INHIBITION_DIMENSION-1))+ HDCELLS(z).*HD_WEIGHT_INHIBITION ;
   end
end
HDCELLS = HDCELLS - hdcells_li_new;


%% global inhibition

for i=1 : HD_DIMENSION_THETA
    if HDCELLS(i) >= HD_GLOBAL_INHIBITION
        HDCELLS(i) = HDCELLS(i) - HD_GLOBAL_INHIBITION;
    else 
        HDCELLS(i) = 0;
    end 
end

%% Global Normalisation
total = 0;
total = sum(HDCELLS);
HDCELLS = HDCELLS./total;

%% Get the current head direction

[value, z] = max(HDCELLS);

z_HDCELLS = zeros(1,HD_DIMENSION_THETA);

z_HDCELLS(HD_AVG_TH_WRAP(z:z+HD_CELLS_TO_AVG*2)) = HDCELLS(HD_AVG_TH_WRAP(z:z+HD_CELLS_TO_AVG*2));

out_theta = mod(atan2(sum(HD_TH_SUM_SIN_LOOKUP *(z_HDCELLS')), sum(HD_TH_SUM_COS_LOOKUP *(z_HDCELLS')))/theta_size, HD_DIMENSION_THETA);


%% create local excitation weight or local inhibition weight

function [weight] = test_create_hdcell_weights(dim,var)

dim_centre = floor(dim/2) + 1;

weight=zeros(1,dim);

for z=1:dim  
    weight(z) = 1/(var*sqrt(2*pi))*exp(-(z-dim_centre)^2)/(2*var^2); 
end

total = sum(weight);
weight = weight./total;       

end