(i.e. in the absence of synaptic interactions) and How could I prove that the neuron did this endogenously?
You will have to add pacamaker currents, such as T-type calcium currents and/or persistent Na currents.
There are several neuron models that fire endogenously. Eg: http://www.scholarpedia.org/article/Plant_model
To probe that the model fire in the absence of synaptic input, I think that the answer is 'to simulate'.
If you tell us the purpose of the model, maybe we can give you a more suitable model for your purposes.
Like Refik is saying, you need to add pacemaker currents. Another good one to try would be Ih. You could base your model on HH equations and try adding the h-current and t-current components.
From a dynamical systems point of view, a pacemaking neuron needs to have a stable limit cycle. You could build such a neuron using the dynamical systems approach, for example by tweaking the computationally efficient Izhikevich model (http://www.izhikevich.org/publications/spikes.htm). Also see Eugene Izhikevich's excellent textbook on the subject: "Dynamical Systems in Neuroscience. The geometry of Excitability and Bursting".
As an additional resource, have a look at the BioModels Database: http://www.ebi.ac.uk/biomodels-main. You will find lots of interesting models to play with, such as this one: http://www.ebi.ac.uk/biomodels-main/static-pages.do?page=ModelMonth%2F2012-03
Cardiac myocytes are a good example of pacemaker cells and arguably the best characterized. These endogenous pacemaker currents will activate and inactivate time- and voltage-dependentl. As long as the cell's membrane potential is at resting level, the cell will fire rhythmic action potentials.
best wishes
Hi,
another nice resource of neuron models may be Opensourcebrain, see links below.
Enjoy exploring :)
Dagmar
http://www.opensourcebrain.org/
http://www.biomedcentral.com/1471-2202/13/S1/O7
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