Carlos von der Becke y Oscar García- Cascades II

d.5. RANKING OF THE 2,3,4 and 5 TERM INTERACTIONS WITH AND WITHOUT STATISTICAL SIGNIFICANCE

1.Table I is the list of the findings, discarding the principal linear effects. It shows the six principal non linear components of a behavioral cascade, three of them triple interactions (less probablea priori ) and three double interactions.

Table with index of significance
Ranked interactions of neurons
Ranking order	Interaction between neurons	Significance index	Conclusion about significance	See Figure
1	x2x4	76	included significant	see Fig 4
2	x1x2	70	included significant	see Fig 5
3	x2x4x5	68	included significant	see Fig 6
4.5	x1x3x4	67	included significant	see Fig 7
4.5	x1x2x3	67	included significant	see Fig 8
6	x4x5	67	included, significant	see Fig 9
7	x1x5	66	excluded, not significant
8	x2x3	65	excluded, not significant
9.5	x1x4x5	64	excluded, not significant
9.5	x1x2x4	64	excluded, not significant
11.5	x2x3x5	63	excluded, not significant
11.5	x3x4	63	excluded, not significant

The highest four-term interaction gives a significance index of 59 (excluded) and the only five-term one, gives 37 (excluded)

2. The six main non-linear two- and three-term interactions are studied on Fig. 4 to 9. At the top is the name of the neurons involved and its sign, either firing or at rest, and some overall statistics. At the middle there is an opened list nested on this particular cell, with the particular frequency (see Table I). At the bottom one sees the 2-D or 3-D representation of the main sequence cycling trajectory.

3. One sees a series of linear and curved arrows under a common title: Main Behavioral Sequence. The arrows indicate the 2 or 3 dimensional excursions among the main sequences behaviors (Fig. 3 with their numbers from (1) to (8), relating changes). Here the analysis points to important conclusions. The order of importance of the comments decrease as the list develops itself. The Fig. 4 comments are stronger than the others, and those of Fig. 9 are comparatively the weakest here considered.

Fig. 4- X₂X₄ MAIN PAIR INTERACTION
             -------------------                      ------------------
             10 FO+ NWC-                              11 FO+ NWC+    
           NONE NON-ZERO                            FOUR NON-ZERO
          COMBINATIONS                             COMBINATIONS
               NO ODOR + TRAPPED1                       ODOR + TRAPPED1                            
               frequency_TOT=0                            frequency_TOT=0.3961                                                                                                                                                                  -------------------                      ------------------






             -------------------                      ------------------
             00 FO- NWC-                              01 FO- NWC+    
           FIVE NON-ZERO                            FOUR NON-ZERO
          COMBINATIONS                             COMBINATIONS
               NO ODOR + UNTRAPPED1                     ODOR + UNTRAPPED1                            
               frequency_TOT=0.3208                       frequency_TOT=0.2830
            -------------------                      ------------------
------------------------------------------------------------------------

             10                                   11
                                                    01010--- 0.0660
                                                    01110--- 0.1509
                                                    11010--- 0.0943
                                                    11110----0.0849
               -----------------                      ___________                              
             00                                   01
                00000---  0.1604                    00010----0.1132
                 00001---- 0.0755                    10010----0.0660 
                   00100---- 0.0566                    00110----0.0755             
                10000---- 0.0189                    10110----0.0283
10100---- 0.0094                   
---------------------------------------------------------------------------

MAIN BEHAVIORAL SEQUENCE


             10                                   11





             00                                   01

4. From Fig. 4: in this 2-D space, there were three cases of nested behaviors, one around 00 with three sequential behaviors, and two others around 01 and 11. No phase transition of simultaneous two digit changes appear. The whole sequence, once the nested behaviors remain hidden, begins with the 00 node, goes to the right and then up, returning back using the same route. The fourth node (10) does not provide any behavior (never the FO neuron can be positive if NWC is at rest, because the links avoid the possibility).
Fig. 5- X₁X₂ SECOND PAIR INTERACTION ------------------- ------------------ 10 NFCL+ FO- 11 NFCL+ FO+ FOUR NON-ZERO TWO NON-ZERO COMBINATIONS COMBINATIONS TRAPPED2 + NO 0DOR TRAPPED2 + ODOR frequency_TOT =0.1226 frequency_TOT=0.1792 ------------------- ------------------ ------------------- ------------------ 00 NFCL- FO- 01 NFCL- FO+ FIVE NON-ZERO TWO NON-ZERO COMBINATIONS COMBINATIONS UNTRAPPED2 + NO ODOR UNTRAPPED2 + 0DOR frequency_TOT=0.4812 frequency_TOT=0.2169 ------------------- ------------------ ----------------------------------------------------------------------------
10 11 10000 ---0.0189 11010-----0.0943 10010----0.0660 11110-----0.0849 10100----0.0094 10110----0.0283
-------------------- ---------------------
00 01 00000 ---0.1604 01010----0.0660 00001----0.0755 01110----0.1509 00010----0.1132 00100----0.0566 00110----0.0755
--------------------------------------------------------------------------- MAIN BEHAVIORAL SEQUENCE
10 11 00 01
5. From Fig. 5: this is the richest map, also in 2-D. The whole behavioral cycle moves counterclockwise around the four nodes. Each node has a strong biological sense. 00 means a free and wandering way of walking not subjected to the rules near a border. 01 means a trapped by the border gait, with no freedom. 11 means behaviors trapped by odor following behaviors which die when entering into the food patch, occasion when the 10 behaviors appear until the exit is marked by outgoing 10 behaviors. This resumes into 00 freedom. On the whole, one concludes that two neurons, NFCL and FO, are enough to detect the principal behaviors. The other three are superabundant, reveiling secondary aspects only.

Fig. 6.- X₂ X₄ X₅ MAIN TRIPLE INTERACTION ------------- ------------- 101 FO+NWC-RBL+ 111 FO+NWC+RBL+ NONE NON-ZERO NONE NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0 -------------- -------------- ---------------- - -------------- 100 FO+NWC-RBL- 110 FO+NWC+RBL- NONE NON-ZERO FOUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 ------------- ------------- ------------- -------------- 001 FO-NWC-RBL+ 011 FO-NWC+RBL+ ONE NON-ZERO FOUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0.O75 freq_TOT=0.3961 -------------- ------------- - ------------ ------------- 000 FO-NWC-RBL- 010 FO-NWC+RBL- NONE NON-ZERO FOUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0.2453 freq_TOT=0.2830 -------------- -------------- ------------------------------------------------------------------------ OPENED LIST 101 111 NONE NONE 100 110 NONE 01010 0.0660 01110 0.1509 11010 0.0945 11110 0.0849 001 011 00001 0.0755 NONE 000 FO-NWC-RBL- 010 FO-NWC+RBL- 00000 0.1604 00010 0.1132 00100 0.0566 00110 0.0755 10000 0.0189 10010 0.0660 10100 0.0094 10110 0.0283 ------------------------------------------------------------------------------------- MAIN BEHAVIORAL SEQUENCE 101 111 100 110 001 011 000 010
6. From Fig. 6: At 3-D space, the main behavioral sequence leaves 4 nodes unused and shows two excursions centered at the 000 node. The first goes back and returns. The other goes right and then up, and returns using the same route. During the second excursion the two nodes visited nest a total of two behaviors each. Both excursions do have biological meaning. The back one marks transitions between a straight an a non-straight free gait. The right & up one, a sequential mix of edge-following, patch-getting-in and patch-getting-out transitions. At the 010 node two separated behaviors unify in one: 00010 trapped by edge-following and 10010 patch leaving with non-edge-following command firing.

Fig. 7- X₁ X₃ X₄ SECOND TRIPLE INTERACTION ------------- ------------- 101 NFCL+SC-NWC+ 111 NFCL+SC+NWC+ NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- -------------- ------------------- -------------- 100 NFCL+SC-NWC- 110 NFCL+SC+NWC- FIVE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0.3208 freq_TOT=0.2830 ------------- ------------- ------------- -------------- 001 NFCL-SC-NWC+ 011 NFCL-SC+NWC+ NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- ------------- - ------------ ------------- 000 NFCL-SC-NWC- 010 NFCL-SC+NWC- NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- -------------- ------------------------------------------------------------------------ OPENED LIST 101 111 10010----0.0660 10110----0.0283 11010----0.0945 11110----0.0849 100 110 10000----0.0189 10100----0.0094 001 011 00010----0.1132 00110----0.0755 01010----0.0660 01110----0.1509 000 010 00000----0.1604 00100----0.0566 00001----0.0755 ------------------------------------------------------------------------------------- MAIN BEHAVIORAL SEQUENCE 101 111 100 110 001 011 000 010
7. From Fig. 7: As happened with Fig. 5, 2-D richest map, here in 3-D a main cycle appears. Again, each used node has biological meaning. Four nodes remain unused. The four main nodes show the route from free walk to trapped by edge-following, trapped by both edge-following and odor, patch-entering, and patch-exiting behaviors, and so again into the first behavior. First back, second right, then making a main diagonal (corresponding to the first important phase transition of the whole analysis, with two of the three digits leaping simultaneously) followed by another diagonal, with exactly the same importance as the one just commented. The two main phase transitions detected are the numbers (6) and (7) of Fig. 3.
Fig. 8.- X₁ X₂ X₃ SECOND TRIPLE INTERACTION ------------- ------------- 101 NFCL+FO-SC+ 111 NFCL+FO+SC+ NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- -------------- ------------------- -------------- 100 NFCL+FO-SC- 110 NFCL+FO+SC- FIVE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0.3208 freq_TOT=0.2830 ------------- ------------- ------------- -------------- 001 NFCL-FO-SC+ 011 NFCL-FO+SC+ NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- ------------- - ------------ ------------- 000 NFCL-FO-SC- 010 NFCL-FO+SC- NONE NON-ZERO NFCLUR NON-ZERO COMBINATIONS COMBINATIONS freq_TOT=0 freq_TOT=0.3961 -------------- -------------- ------------------------------------------------------------------------ OPENED LIST 101 111 10100 ----0.0094 11110 ----0.0849 10110 ----0.0283 100 110 10000---- 0.0189 11010--- 0.0945 10010---- 0.0660 001 011 00100 ----0.0566 01110 ----0.1509 00110 ----0.0755 000 010 00000--- 0.1604 01010--- 0.0660 00001---- 0.0755 00010---- 0.1132 ------------------------------------------------------------------------------------- MAIN BEHAVIORAL SEQUENCE 101 111 100 110 001 011 000 010
8. From Fig. 8: Although the biological sense of the nested nodes differs somewhat from the preceding case, this figure also shows a cyclic non- repeating route beginning and ending at 000. The two important phase transitions are now (6) and (8), instead of (6) and (7), showing clearly that step (6) in Fig. 3, is the paradigmatic behavioral phase transition of the whole sequence.

Fig 9.- X₄ X₅THIRD PAIR INTERACTION ------------------- ------------------ 10 NWC+ RBL- 11 NWC+ RBL+ EIGHT NON-ZERO NONE NON-ZERO COMBINATIONS COMBINATIONS TRAPPED + NO FREE TURN TRAPPED + FREE TURN freq_TOT=0.6791 freq_TOT=0 ------------------- ------------------ ------------------- ------------------ 00 NWC- RBL- 01 NWC- RBL+ FIVE NON-ZERO FOUR NON-ZERO COMBINATIONS COMBINATIONS UNTRAPPED + NO FREE TURN UNTRAPPED + FREE TURN freq_TOT=0.2453 freq_TOT=0.0755 ------------------- ------------------ ---------------------------------------------------------------------------- 10 11 00010----0.1132 none 00110----0.0755 01010----0.0660 01110----0.1509 10010----0.0660 10110----0.0283 11010----0.0943 11110----0.0849 _________________ _________________ 00 01 00000 ---0.1604 00001----0.0755 00100----0.0566 10000----0.0189 10100----0.0094 ------------------------------------------------------------------------- MAIN BEHAVIORAL SEQUENCE 10 11 00 01
9. From Fig. 9: An L shaped trajectory in 2-D first goes right and back and then goes up and back. The multiple behavior-nested 10 node is the only novelty shown. It marks a very long subcascade of behaviors.
d.6 MORE ON BEHAVIORAL SUBCASCADES AND SUB-SUB-CASCADES.
1. The analysis just ended could be rounded up in a proposal for important subcascades shown. The authors deduce the following result's interpretation. The Fig. 5, studying the X1X2 interaction, is the best graphical route for the whole behavioral cascade. The main subcascade goes from (8),(1) to (5) , and means freedom of food patch effects. Now, when the Fig. 4 is compared to the previous proposal, one finds that inside the (1) to (5) subcascade there are two distinct sub-sub-cascades. Those two are (8),(1),(2),(3) and the (3),(4),(5). The first is untrapped and the second is edge-trapped. The same Fig. 4 shows still another subcascade, that is, (5),(6),(7), an important part of Fig. 5. The biological meaning is that the subcascade nests and hides the inside-patch behaviors. As mentioned before, there is an arousal-to-exit sub-sub-cascade sometimes striving inside of the (5),(6),(7) subcascade (see d.4.3).
2. One should then say that the search for mesoscopic and microscopic behavorial building blocks, that are nested and hidden inside of macroscopic behavior cascades, can go as deeply as one desires, the limit being the number of distinct functional neurons mapped. With only two neurons mapped, a very small number of cascades and subcascades is possible, but no sub-sub-cascade whatsoever. And that's that. This seems a general neuroethological rule.

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