2. The biophysics of Purkinje computation and coding
The sodium-potassium pump controls the intrinsic firing of the cerebellar Purkinje neuron
Michael Forrest1* Mark Wall2 Daniel Press2
1Department of Computer Science and 2Department of Biological Sciences,
University of Warwick,
Coventry, CV4 7AL, UK

3. The equations and circuit diagram above are summarised in the H-H membrane equation, that describes a capacitance and three ionic currents in a patch of membrane

4. The sodium and potassium conductances are voltage dependent !
Hodgkin and Huxley’s phenomenological mechanism to account for this voltage dependence:
conductances result from assembly of fictional, independent, charged, ‘gating particles’.
Each particle has two possible locations, on either side of the membrane, and their location is voltage and time dependent.
For each gating particle, one location is conducive to ion flow and the other is not.
For, a current to flow all that current’s particles must be in the permissive location.
So called activation particles are those that tend to be in the permissive state with depolarisation, whilst inactivation particles are those that tend to be in the non-permissive state with depolarisation.
“Certain features of our equations [are] capable of physical interpretation, but the success of our equations is no evidence in favour of the mechanism of permeability change that we tentatively had in mind when formulating them.”
The potassium conductance was modelled with four activation particles (the n term)
The sodium conductance was modelled with three activation particles (the m term) and a single inactivation particle (the h term)

5. The modern interpretation of the Hodgkin-Huxley model
Voltage gated ion channels are transmembrane protein complexes containing a pore permeable to one or more ions, gated in an all-or-none fashion, according to the membrane potential
The n, m and h particles in the H-H model are now popularly re-interpreted as voltage and time dependent, two-state ion channel components driving the workings of two-state gates in the channel pore.
INa+
IK+

7. The cerebellar connectivity motif
The cerebellum
The Na+/K+ pump pump is electrogenic

8. Purkinje cells can intrinsically fire action potentials in a repeating trimodal or bimodal pattern. The trimodal pattern consists of tonic spiking, bursting and quiescence. The bimodal pattern consists of tonic spiking and quiescence. It is unclear how these firing patterns are generated and what determines which firing pattern is selected. We have constructed a realistic biophysical Purkinje cell model that can replicate these patterns and which shows the Na+/K+ pump to set the Purkinje cell operating mode. This model has experimental validation in its ability to replicate presented in vitro Purkinje cell recordings in the presence of ouabain, which irreversibly blocks the Na+/K+ pump. We show that the Na+/K+ pump controls intrinsic Purkinje firing and propose that Na+/K+ pumping is modulated to switch the Purkinje cell between different firing modes in a physiological setting. On this basis, we hypothesise that the Na+/K+ pump is a computational element in Purkinje information coding.

9. Experimental data. Trimodal pattern
Model output. The model can replicate the trimodal pattern of firing.

10. Experimental data
Model output

11. In vivo experimental data [Lowenstein et al., 2005]
Model output

12. Conclusion
How does the nervous system compute and what is the neural code? Presently, we know little and the answer is the elusive holy grail of brain research. In the classical postulate, the firing rate of neurons is considered to be the significant variable in neural computation. However, it is becoming increasingly clear that this view is overly simplistic. For example, in our work we reach the conclusion that Purkinje neurons encode information not just in their spiking but in the patterning and length of quiescent periods, which are in turn dictated by computations performed by intracellular ion systems (Na+ and Ca2+). Although we have only studied Purkinje neurons, there is the possibility that this finding might be more general and could relate to the neuronal dynamics of other neural units.