There are too many papers that present a new spike-train probability model. Is there any book or paper that introduce and compares them? What is the best model to generate spike trains mathematically?
There are numerous treatments, as this is a central question in computational neuroscience. For example:
Grün, S., & Rotter, S. (2010). Analysis of parallel spike trains (Springer Series in Computational Neuroscience Vol. 7)
Doya, K. (Ed.). (2007). Bayesian brain: Probabilistic approaches to neural coding (Computational Neuroscience).
Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models: Single neurons, populations, plasticity. Cambridge university press.
Izhikevich, E. M. (2007). Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience)
Ermentrout, G. B., & Terman, D. H. (2010). Mathematical foundations of neuroscience (Interdisciplinary Applied Mathematics Vol. 35)
These are just a few of some of the general treatments of approaches to spike-train modelling. There is no best model, because "all models are wrong". In general, the ability to fully model the basic functions of any cell in any living system is beyond us. Naturally, single neuron models attempt to model more completely the intracellular dynamics that (among other things) govern the generation of action potentials as well as inhibits these. However, this level of precision is neither particularly useful not well suited to neural population models. Firing rates and just about everything else are always estimations, and how precise one requires these estimates to be depends upon what one is attempting to model. The kind of probabilistic approaches I'd use for single neuron models is practically unrelated to that I'd use for neuronal network models. Think of it this way: virtually every single model of neural electrophysiology relies on classical electrodynamics. Classical electrodynamics is wrong. So why doesn't everybody model cellular dynamics using quantum physics (or quantum field theory or quantum electrodynamics)? Because so far as we can tell, the difference between the results obtained by using quantum physics at the cellular level is usually negligible (so far as I know, experiments in quantum coherence haven't resulted in the creation of a superposition state for systems larger than 430 atoms, meaning that even in artificially created environments quantum effects aren't found at scales as large as a single neuron). However, it is vastly more complicated if it is even possible to create a quantum mechanical model of neuronal dynamics. So, we use classical theories we know can't be right. Why? Because all models are wrong, but some are useful.
Thank you Andrew. It was useful. I am thinking about your comment.
Fred Rieke, David Warland, Rob de de Ruyter van Steveninck, William Bialek, Spikes: Exploring the Neural Code (Computational Neuroscience), MIT press.
I agree with Andrew 'all models are wrong' and 'there is no best model'
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