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Figure 2 | BMC Neuroscience

Figure 2

From: Cross-approximate entropy of cortical local field potentials quantifies effects of anesthesia - a pilot study in rats

Figure 2

Calculation of XapEn. Short excerpt of raw data from three channels shown to illustrate the computation of XApEn. Parameters were set to r = 23 μV (tolerance) and m = 1 (sequence length). Thus, the 'sequence' length is one data point. For illustration purposes we choose data point i = 10 of Channel 1. Similar sequences (data points of similar amplitude) of Channels 2 and 3 are found within the upper rectangle representing tolerance r, i.e. data points 6 and 7 of Channel 2 and data point 15 of Channel 3. This leads to coefficients C 10 1 ( r ) ( C h a n n e l 1 | | C h a n n e l 2 ) = 2 25 1 + 1 = 2 25 and C 10 1 ( r ) ( C h a n n e l 1 | | C h a n n e l 3 ) = 1 25 . In the second pass, sequence length is extended to m+1 = 2 points so that now the two point-sequence Channel 1 (10,11) is under consideration. For all previously identified data points in channels 2 and 3 (namely, those similar to Channel 1 (10)) the algorithm checks if the consecutive data point is similar to Channel 1 (11) and hence a pattern similar to Channel 1 (10, 11) exists. Graphically this means that two consecutive data points of channels 2 and 3 have to reside within the upper and lower rectangle, respectively. Only Channel 2 (7, 8) fulfils this requirement. This leads to coefficients C 10 1 + 1 ( r ) ( C h a n n e l 1 | | C h a n n e l 2 ) = 1 25 and C 10 1 + 1 ( r ) ( C h a n n e l 1 | | C h a n n e l 3 ) = 0 25 .

This pattern matching procedure is performed for all possible sequences so that: X A p E n ( C h a n n e l 1 | | C h a n n e l 2 ) = ( n m + 1 ) 1 i ln C i m ( C h a n n e l 1 | | C h a n n e l 2 ) ln C i m + 1 ( C h a n n e l 1 | | C h a n n e l 2 )

If a specific pattern in one channel can not be detected in the other, as in the example above, the corresponding ln C i m ( x | | y ) is undefined. In the analyzed data sets and with the parameters chosen, this case did not occur.

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