Analysis of Family Structures Reveals Robustness or Sensitivity of Bursting Activity to Parameter Variations in a Half-Center Oscillator (HCO) Model

3D view of the 99,066 realistic HCO instances. Plot of all rHCO instances projected onto the 3D space given by the maximal conductances of I_P, I_K2, and I_Leak. Each point displays a pie chart of the E_leak of all instances from the group having the same values of ḡ_P, ḡ_K2, and ḡ_Leak as the respective projected point. The number of instances projected onto each point in the 3D space is shown by the size of the E_leak pie (as radius). For a better visualization of the points, we applied the natural logarithm to the sizes of the pies.

The number of instances within the rHCO group and fHCO group vs their period values for each value of ḡ_h. fHCOs are in blue and rHCOs are in red. Period values were between 1 and 40 s for the fHCOs and between 5 and 15 s for the rHCOs. Plots were obtained using the bar Matlab function (R2014a). Each bar shows the period values within one unit or between [n, n + 1). The bars corresponding to the realistic instances were shifted with 0.3 on the x-axis for better visualization. ḡ_h values are given as the percentage of the canonical (100%).

Distribution of h families of rHCOs based on the sequence breakdown of their members. Labels at the margins indicate the beginning (vertical dimension) and end (horizontal dimension)ḡ_h values of the family sequences. Colors indicate the number of members within the family. The titles of panels show the total number of families [total number of interrupted families plus the number of noninterrupted families (star)]. Numbers under each panel title give the total number of interrupted families (left) and the number of noninterrupted families (right). For each panel, the y-axis indicates the number of families. Bins corresponding to the families having all members with noninterruptedḡ_h values are marked with an orange star. All panels show data at the same scale.

Distribution of CaS families of realistic HCOs based on the sequence breakdown of their members. Labels at the margins indicate the beginning (vertical dimension) and end (horizontal dimension)ḡ_CaS values of the family sequences. Colors indicate the number of members within the family. The titles of panels show the total number of families [total number of interrupted families plus the number of noninterrupted families (star)]. The numbers under each panel title give the total number of interrupted families (left) and the number of noninterrupted families (right). For each panel, the y-axis indicates the number of families. Bins corresponding to the families having all members with noninterruptedḡ_CaS values are marked with an orange star. Empty panels have been removed. To visualize bins with few families, panels are at different scales.

Distribution of missing members from h families of rHCOs for eachḡ_h value. The activity type of the missing members is color coded as shown in the legend.

Distribution of missing members from P families of rHCOs for each ḡ_P value. The activity type of the missing members is color coded as shown in the legend.

Variations of burst characteristics (period and spike frequency) vs ḡ_h for the realistic HCOs. Lines connect adjacent family members. A, Period vs ḡ_h for h families with eight members (144 families). All curves are monotonically decreasing. B, Period vs ḡ_h for h families with four members (4,322 families). For four-member families, regardless of the beginning point, all curves are monotonically decreasing. C, Spike frequency vs ḡ_h for h families with eight members. Most of the curves are monotonically increasing. D, Spike frequency vs ḡ_h for h families with four members. For four-member families, regardless of the beginning point, the vast majority of curves are monotonically increasing.

Sensitivity of period to the variation of ḡ_h for h families of realistic HCOs. A, Plots of period vs ḡ_h for h families with eight members (144 families). Lines connect adjacent members of an h family forming a curve. Curves were split into three groups according to their period sensitivity (slopes), as follows: high (steep slopes; 1); medium (medium slopes; 2); and low (shallow slopes; 3). B, Parameter values vs the number of distinct families for all h families with eight members separated into these three groups of sensitivity. The size of each point quantifies how many different families have that parameter value (here, we scale up the sizes by five times for a better display). Black diamond shapes show the median values for each parameter. C, Parameter values vs the number of distinct families from all h families with four members separated into the three groups of sensitivity. For simplicity, plots do not show data for ḡ_SynG since the graded transmission did not have a significant influence on burst characteristics (see Sensitivity of period to variations of ḡ's, and Sensitivity of spike frequency to variations of ḡ's).