Figure 1.
Leech heart motor neuron circuit in segments 3–12 and input/output pattern.
A. Simplified circuit diagram for heart (HE) motor neurons depicting the premotor heart (HN) interneurons (ganglia of origin indicated) and the synapses from the former to the latter. Adjacent to each neuron is a representative extracellular recording with the middle spike indicated by a small symbol above each burst. One period is indicated by the grey background bar. Connectivity between interneurons is not shown (for detail, see [20], [57], [58]). Note that ipsilateral midbody motor neurons (e.g., HE(8) and HE(12) above, highlighted with the red broken line) receive the same complement of inputs. B. Relative phasing of first, middle and last spikes in heart motor neurons and interneurons recorded from a single animal, a portion of which is shown in panel A, as reported by Norris [23] and used previously in earlier modeling efforts [9], [10]. Error bars indicate standard deviations. The peristaltic pattern exhibits a strong rear-to-front phase progression in both the interneurons and motor neurons, whereas the synchronous pattern exhibits a minimal phase progression. Note that the HE(8) synchronous (s) and peristaltic (p) motor neuron bursts are nearly perfectly out of phase, unlike the HE(12)p and HE(12)s motor neuron bursts which partially overlap. C. Relative synaptic strength of synapses onto heart motor neurons calculated from spike-triggered averaged IPSCs. Adapted with permission from [23].
Figure 2.
A pair of electrically coupled 7-compartment model heart motor neurons with their inhibitory inputs shown. The inputs to each model motor neuron were the prerecorded input spike times from the premotor interneurons (delayed based on the motor neuron pair’s ganglion of origin), and the relative strength of the synapses. The soma, neurite and axon compartments contained active conductances in addition to the passive membrane capacitance and leak conductance, whereas the secondary neurite and synaptic compartments were only passive. The soma compartment contained a gK1 and gK2; the neurite compartments (Neurite 1, Neurite 2, Neurite 3) each contained gK1, gK2, gKA, gP, gCaS, and gKCa; and the axon compartment contained gNa, gK1, gK2, and gKA. The synaptic compartment contained synaptic elements and was electrically coupled (gcoup) to the contralateral heart motor neuron. An instance of the abstract model was defined by specific values for the maximal conductance () density of each conductance shown above.
Figure 3.
Fitness metrics used to quantitatively evaluate model output.
The soma compartment membrane voltage was split into low pass and high pass components, spikes were identified, and 5 fitness metrics were calculated: phase, duty cycle, mean intraburst spike frequency, mean spike height, and slow-wave height. A. The unfiltered soma compartment membrane voltage (blue trace) with the first spike (fuchsia vertical line) and last spike (red vertical line), phase reference (green vertical lines), phase target (black circle and ochre line), and measured phase (blue dot and topaz line), burst duration (black horizontal line), and period (red horizontal line) indicated. The low-pass trace shown in panel C is in red behind the primary trace. The relative phase of the middle spike was calculated with reference to the middle spike of the reference HN(4)p interneuron as shown in Figure 1. The duty cycle was calculated from the phase of the last spike minus the phase of the first spike. All phases were measured with reference to the HN(4)p middle spike. B. High-pass filtered soma compartment membrane voltage (blue trace) with identified spikes (red vertical lines) and spike height indicated. Spike frequency is the inverse of the inter-spike interval within the identified burst. The spike height was defined as the value of the high pass trace at the peak of each spike within the burst. Both the spike height and spike frequency were averaged within each burst. C. Low-pass filtered soma compartment membrane voltage (black trace) with slow-wave height defined as filtered voltage difference at the middle spike relative to the minimum voltage in the inter-burst region.
Table 1.
Dimensions and upper bounds of conductance densities allowed in the MOEA.
Table 2.
Hodgkin-Huxley style membrane conductance formulae.
Figure 4.
Steady state activation and inactivation curves for voltage-gated conductances.
The activation and inactivation gating variables for each of the 7 active membrane conductances used are shown. Solid lines represent activation curves and the dotted lines represent inactivation curves. All conductances except the calcium sensitive potassium conductance have been used in previous heartbeat system models [8], [57].
Table 3.
Voltage-gated conductance model parameter values.
Table 4.
Targets, error thresholds and the mean, minimum, maximum, and standard deviation of set C fitness values for each fitness metric.
Figure 5.
Proportions of model instances, depicted as Venn diagrams, falling within target ranges of fitness metrics.
Out of a total of 734,205 model instances which were simulated, 490,023 were at least quasi-functional. The left hand column of Venn diagrams shows the model instances which were capable of producing not only basic fitness metrics (average spike frequency, spike height, slow-wave height) but also the proper phase for the peristaltic or synchronous mode in the indicated heart motor neuron. For example, there were 54,361 models which produced a phase and duty cycle within an acceptable deviation from the target for the HE(8) motor neuron with the peristaltic input pattern, but only 4,471 of these also produced the correct activity with the synchronous input pattern. 431 instances produced output which was within the target range for all metrics, i.e. they produced the target output with all four input patterns: HE(8)p, HE(8)s, HE(12)p and HE(12)s. Three sets were used in subsequent analyses based on the targets they achieved in addition to the basic metrics: set A (blue), the instances which achieved output within the target range for the HE(12) metrics; set B (green), HE(8) metrics; and set C (red), the intersection of sets A and B.
Figure 6.
Membrane currents and soma membrane potential in the neurite 2 compartment of two model instances.
All actively gated currents (IK1, IK2, IKA, IP, ICaS, IKCa) in the neurite 2 compartment from two representative HE(12)p model instances. Traces are offset for clarity and the solid black line indicates the zero reference for the corresponding trace. On the soma voltage trace, small red circles indicate identified spikes, large red dots indicate the phase reference, blue circles indicates target phase,the short vertical black lines indicate measured phase and the horizontal black line indicates the target range. A. Representative model instance from set C with low K2 and
P but
KA in the middle of the allowable range. B. Extreme model instance selected from set C with high
K2, near maximum
P for set C, and low
KA.
Figure 7.
Effect of parameter interaction on fitness set.
Functional model instances projected onto 2-D parameter space for each pair of parameters are shown for sets A (blue), B (green), and C (red). Each subplot is a layered scatter plot of set C over set B over set A of the parameter values of functional model instances for each pair of parameters. Each point may represent many models. Axes are from the minimum to maximum allowable value for the parameter indicated. No clear relationship between pairs of parameters was obvious upon inspection in most cases, but some did present interesting structures. The highlighted subplots show three particularly interesting relationships that are apparent upon inspection (shown as insets above right): K2 appears correlated with
P and is somewhat restricted in its distribution;
KCa and
CaS appear to form a non-linear relationship; and electrical coupling (
coup) is generally restricted to lower values in HE(8) motor neurons (set B) than in HE(12) motor neurons (set A). Black tick marks on the axes and boxes on highlighted subplots indicate the baseline model’s parameter value.
Figure 8.
Partial Correlation (ρ) matrix for a subset of parameters.
Warmer colors indicate positive and colder colors negative partial correlations. In general, pairs of conductances which oppose each other were positively correlated and pairs that could compensate for each other were negatively correlated. Numbers shown are calculated ρ. All partial correlation shown were significant (p<0.00032).
Figure 9.
Phase, duty cycle and spike frequency sensitivity to neurite P parameter perturbation.
Maximal conductance parameters were perturbed by ±50% and 25% of their initial value for all model instances in set C. The resulting changes in phase, duty cycle and average spike frequency are plotted above for each mode of HE(8) and HE(12) motor neurons. Data shown as mean ± std.
Figure 10.
Phase, duty cycle and spike frequency sensitivity to neurite KA and
K2 parameter perturbation.
Maximal conductance parameters were perturbed by ±50% and 25% of their initial value for all model instances in set C. The resulting changes in phase, duty cycle and average spike frequency are plotted above for each mode of HE(8) and HE(12) motor neurons. Data shown as mean ± std.
Figure 11.
Last and first spike time during F/I ramp injection.
First spike vs. last spike relative to the peak hyperpolarizing current injection is shown as scatter and box plots for model instances from subsets A B and set C. Median, 75th and 25th percentile indicated by the center line and edges, respectively, for each box. Red crosses indicate outliers. Model instances were probed with a 5s triangular ramp current from 0 to -0.5 nA and back to 0 injected into the soma compartment. There was a statistically significant difference in last and first spike time for the FI protocol based on set, F(6, 2782) = 624.171, p<0.0005; Pillai’s trace = 1.148, partial η2 = 0.574. Post hoc tests (Bonferroni) showed a significant difference between set C and subset A (p<0.0005) as well as between set C and subset B (p<0.0005) for the first spike time, but not between subsets A and B (p = 0.111). There was a significant difference between all sets/subsets for the last spike time (p<0.0005).