We created a novel reduction method of a neuronal dendritic morphology that we have termed preserved path reduction or PPR. In the PPR model, an electrical path between the soma and a particular spine on a remote dendrite is preserved as an explicit, unreduced path, whereas the rest of the morphology is reduced to a relatively small number of equivalent cylindrical compartments. The axial resistivity (Ra), input resistance (Rin), and the membrane time constant (τ) remained unchanged, while the membrane resistance (Rm) and capacitance (Cm) were adjusted based on the difference in surface area between the actual and reduced compartments. This approach creates a good fit between the full and reduced passive models. In addition, active conductances were scaled in the equivalent cylinders, as were the fluxes for calcium flowing into the equivalent cylinders through voltage-gated calcium channels on the membrane. The resulting reduced Purkinje cell model can be used in single cell simulations or in network simulations. Use in either scenario will yield increased computational efficiency, decreased run time, and greater feasibility for large networks. Further, we showed that multiple equivalent and unreduced cylinders can be connected in Virtual Cell to model the neuron as an electrical circuit. Results for action potentials and E.P.S.P.s at the spine and the soma in Virtual Cell closely match those in NEURON. Implementing the PPR model motivated the development of new approaches in Virtual Cell that facilitate the investigation of multicompartmental electrophysiological and biochemical processes. Specifically, we were able to mimic a NEURON model by specifying ionic and molecular fluxes between multiple compartments in Virtual Cell. As a result, it is now possible to create models of cellular geometries that consist of “virtually connected” conductors that each have an equipotential membrane—similar to the NEURON modeling paradigm. This will also allow users to take advantage of the power of Virtual Cell to add and easily edit detailed biochemistry combined with the electrophysiology model. In addition, this work led to performance optimization in Virtual Cell that will be beneficial to users with large models, particularly ones designed to investigate neuronal electrophysiology. This model contains approximately 350 variables and 2000 functions, and is one of the larger compartmental models publicly available in the Virtual Cell database.
However, we wish to emphasize that Virtual Cell is not intended as a tool for modeling electrophysiology in complex neuronal morphologies. The PPR method allows us to simplify a complicated neuronal geometry in order to create models more tractable in Virtual Cell. Reproducing the full Purkinje neuron geometry as either a multicompartment ODE model or a full spatial PDE model in Virtual Cell would be completely impractical, but for different reasons. The ODE model cannot be constructed with the graphical user interface because it is not designed to specify the hundreds of compartments required to represent complex neuronal morphologies. The PDE spatial model would actually be easier to build in Virtual Cell, because the experimentally derived geometry can be directly mapped to the physiology; but the PDE solvers in Virtual Cell would require too much memory and too much computational time to deal with the high spatial resolution required to simulate the full Purkinje geometry. For these reasons, the full Rallpack cable test (Bhalla et al. 1992
) also cannot be implemented in Virtual Cell. However, the Purkinje PPR model in Virtual Cell was validated against the implementation in NEURON. The correspondence of the simulation results on the two platforms for the PPR model (), with its 17 heterogeneous compartments each with active channel conductances, might be considered an even more rigorous validation than the more idealized Rallpacks models.
We developed this new approach to neuronal morphology reduction, because of our desire to have a computationally efficient Purkinje cell model in which we could investigate issues that require combined detailed biochemistry and electrophysiology. None of the available reduction methods was suitable for our study. The cerebellar Purkinje neuron is highly and asymmetrically branched, and does not obey the “3/2 rule”. Further, we needed to preserve a path from a single spine to the soma, in order to appropriately compare simulations to experiments. At the same time, we did not wish to maintain explicit dendritic arbors that branch off of the explicit path, as that would not be computationally feasible in VCell, particularly once biochemistry was added.
Rall’s and Rinzel’s traditional methods for global reduction assume, among other things, that the sum of the 3/2 power of the daughter dendrite diameters is equivalent to the parent dendrite diameter raised to the 3/2 power (Rall and Agmon-Snir 1998
). These methods also assume that terminal branches are at the same electrotonic distance from the soma. The terminal daughter branches can be lumped into equivalent cylinders. This process can be continued iteratively until the whole dendritic tree is represented by a few—or even a single—equivalent cylinders (Rall and Agmon-Snir 1998
). However, most realistic neurons do not satisfy these assumptions (Rall and Agmon-Snir 1998
). As a result, a few laboratories have suggested alternative requirements for reduction. Douglas and Martin preserve dendritic surface area as a function of electrotonic distance from the soma (Douglas and Martin 1993
). However, mapping dendrites to equivalent cylinders based on their electrotonic distance from the soma is not appropriate in the preserved path reduction method, since dendrites are lumped together based on branch points from the explicit path.
Two candidate algorithms were ultimately considered: the Bush-inspired method (see Reduction Methods
in the Supplementary Material
) and the PPR model. Using the radius and length equations presented by Bush and Sejnowski (Bush and Sejnowski 1993
) in various approaches to preserved path reduction resulted in the creation of the Bush-inspired model. This reduced model used the scaling factor 4.5 to adjust the membrane resistance and membrane capacitance, as well as voltage-gated ion channel conductances. This scaling factor was determined iteratively by adjusting Rm
in the reduced model fit Rin
in the full model. This phenomenologically derived scaling factor is used globally for all compartments, and does not depend on the particular geometry or properties of one compartment or the group of individual dendrites that it represents. The resting potential of the Purkinje neuron in this simplified model was hyperpolarized relative to the full model. Further, passive and active properties were different between the Bush-inspired model and the full model, when voltage-gated ion channels were added to the dendrites in each model. This led us to create the PPR model using a scaling factor that is theoretically derived by taking into account the differences in surface area between the full and reduced models and by appropriately factoring in the series versus parallel connectivity of the highly branched Purkinje neuron dendritic arbor. Moreover, this scaling factor is separately calculated for each individual compartment in the PPR model. Each reduced compartment has its own scaling factor, based on its unique decrease in surface area relative to the group of dendrites it represents. Thus, unlike in the Bush-inspired model, Rm
are not phenomenologically derived, and are scaled individually in the equivalent cylinders, not globally over the entire geometry. The resting membrane potential in the PPR model is similar to that of the full model. Passive and active properties also closely fit those of the full model. The PPR model overall reproduces the behavior of the full model.
In conclusion, this study reports a novel reduction method that can be used to simplify complex neuronal geometries. Conditions that encourage using the reduction method developed in this study for any neuron with complex geometry include (i) dendrites that contain voltage-gated ion channels (as opposed to passive reduction methods that ignore active cases), (ii) a high degree of asymmetrical branching, (iii) a great number of dendrites in series as well as in parallel, (iv) significant branching off of an explicit path of interest yielding a large number of compartments. The PPR model enables us to simultaneously meet three challenges in modeling neuronal cell biology. First, the Purkinje neuron geometry is complicated, with large degrees of series and parallel branching, such that other reduction methods that we considered were deemed unsuitable to accurately reproduce the electrophysiology of the full geometry. Second, we can model both explicit spine and soma physiology in the same model, which—to the best of our knowledge—has not previously been successfully attempted; this requirement precludes the use of global, restricted, or dynamic reduction methods. Third, the PPR reduction scheme is tractable enough to be transferred to a modeling software system, Virtual Cell, which can efficiently simulate complex biochemistry added to the model. Thus, the PPR model can be used to investigate interactions between biochemistry and electrophysiology. Studies are now underway involving the application of the PPR model to normal physiology and pathophysiology of cerebellar Purkinje neurons, with particular attention to cerebellar dysfunction in mice and humans.