Figure 1.
Class containment diagram for a hybrid model in PyDSTool.
represent ‘internal’ model interface objects that wrap
sub-models (
). If global consistency conditions are applied to these sub-models, then
‘external’ model interface objects,
, may also be provided. Each iMI may either contain a non-hybrid or another hybrid model object (an example is shown). Non-hybrid models combine with a GeneratorInterface to make a thin wrapper for Generator objects, ensuring API-compatibility with hybrid model objects and other MIs and thus promoting interchangeability. Conditions in each eMI specify a target combination of truth or falsity of one or more constituent features. The features measure properties of the corresponding iMI and compare them to those in some external data such as a user-imposed logical template or experimental data.
Figure 2.
Structure of a ModelInterface (MI) class.
An MI wraps a hybrid or non-hybrid model, providing users an option to add pre- and post-simulation code to validate input and output, or otherwise filter or transform the data flow. The control structure also permits failure recovery in the model to be added.
Figure 3.
Schematic of model reduction methodology with hybrid systems using spatial decomposition.
A) The complex model involves many inter-dependent state variables (black boxes), depicted in a connectivity graph. Analysis of a particular model behavior indicates that some inter-dependencies are effectively weak (dashed lines), suggesting a functional decomposition into sub-model components. One such sub-model is highlighted by the yellow oval. B) The internal dynamics of the sub-model is analyzed in the context of known input and output conditions, and a reduced model of the dynamics is derived that closely mimics the original sub-model. The puzzle piece indicates that the reduced model is derived under certain explicit constraints and assumptions that relate to the broader context of the original model. C) Further testing of the reduced component involves embedding it back into the full system as a surrogate for the original sub-model, possibly fine-tuning reduced model parameters to maximize the overall model output similarity under various conditions. If successful, the reduced component represents an abstracted description of the mechanism of that part of the model under these conditions. D) This process can be repeated for other sub-models, building a global hypothesis of the whole mechanism.
Figure 4.
Schematic of the temporal aspects of model reduction with hybrid systems.
This example assumes a model with hub-like connectivity, exhibiting multiple scale dynamics, and a periodic behavior (period ), but a similar process can be described for non-periodic dynamics. State variables are shown by boxes and their inter-coupling by lines. A) Dominant scale analysis identifies Regime I over some time window
(indicated on the blue time axis) within which a subset of the variables (yellow oval) are the most influential on the system's output; the other connections are effectively weak (dashed lines). B) The internal dynamics of the resulting sub-model for the regime (yellow puzzle piece) is analyzed in the context of known input and output conditions alongside the full model under equivalent conditions, and the parameters and contextual conditions for the reduction are tuned to maximize the accuracy of this representation over
. C) The consistency of the sub-model with the full dynamics beyond
is tested for the generation of accurate cyclic behavior over a period
for
, allowing for further refinement. D) The process in A–C is repeated for other regimes, creating four consecutive sub-models in this example. These should form a self-consistent cycle of entry and exit conditions (indicated by matching puzzle pieces) such that from the composition emerges a periodic behavior closely matching that of the full model.
Figure 5.
The graph compares the period of regular AP firing in the original Hodgkin-Huxley model (‘x’ markers) and the hybrid reduction (‘o’ markers) as a function of applied current . Zero period indicates no APs (steady state).
Figure 6.
Voltage trace fits for various hybrid AP models.
Voltage traces versus time summarize the qualitative fit of four smooth biophysical models of APs (solid lines) with their hybrid counterparts (dashed lines). The four sub-model regimes of the AP are indicated with Roman numerals, with onsets indicated by solid square or circular markers. A) The fast interneuron model studied here. B) Original Hodgkin-Huxley parameters. C) A Wang-Buzsáki form of interneuron. D) A heart interneuron model with a larger set of sub-threshold and AP ionic currents. Adapted from Figure 7 of Clewley [46], in which full model and analysis details can be found.