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First, we examine whether generalized linear models can reproduce a suite of canonical spiking behaviors exhibited by the well-known Izhikevich model (Izhikevich, 2003, 2004). The Izhikevich model is a biophysically inspired model of intracellular membrane potential defined by a two-variable system of ordinary differential equations governing membrane potential and a recovery variable :
formula
2.1
formula
2.2
with spiking and voltage reset governed by the boundary condition
formula
2.3
where is injected current, denotes the next time step after , and parameters determine the model's dynamics. Different settings of these parameters lead to qualitatively different spiking behaviors, as shown in Izhikevich (2004). We focus on this model because of its demonstrated ability to produce a wide range of response properties exhibited by real neurons. (For the parameter values used in this study, see Table 1 in the appendix; additional simulation details are also in the appendix.)
Table 1:
Parameters of the Izhikevich Neuron for Dynamic Behaviors Shown in Figures 2–6, 8–9, and 11.
neuron typeabcdIdt (ms)
Tonic spiking 0.02 0.2 65 14 0.1 
Phasic spiking 0.02 0.25 65 0.5 0.1 
Tonic bursting 0.02 0.2 50 10 0.1 
Phasic bursting 0.02 0.25 55 0.05 0.6 0.1 
Mixed mode 0.02 0.2 55 10 0.1 
Spike frequency adaptation 0.01 0.2 65 5 20 0.1 
Type I 0.02 0.1 55 25 0.01 
Type II 0.2 0.26 65 0.5 0.01 
Spike latency 0.02 0.2 65 3.49 0.1 
Resonator 0.1 0.26 60 0.3 0.5 
Integrator 0.02 0.1  27.4 0.5 
Rebound spike 0.03 0.25 60 0.1 
Rebound burst 0.03 0.25 52 0.1 
Threshold variability 0.03 0.25 60 2.3 
Bistability I 1.5 60 26.1 0.05 
Bistability II 1.5 60 26.1 0.05 
neuron typeabcdIdt (ms)
Tonic spiking 0.02 0.2 65 14 0.1 
Phasic spiking 0.02 0.25 65 0.5 0.1 
Tonic bursting 0.02 0.2 50 10 0.1 
Phasic bursting 0.02 0.25 55 0.05 0.6 0.1 
Mixed mode 0.02 0.2 55 10 0.1 
Spike frequency adaptation 0.01 0.2 65 5 20 0.1 
Type I 0.02 0.1 55 25 0.01 
Type II 0.2 0.26 65 0.5 0.01 
Spike latency 0.02 0.2 65 3.49 0.1 
Resonator 0.1 0.26 60 0.3 0.5 
Integrator 0.02 0.1  27.4 0.5 
Rebound spike 0.03 0.25 60 0.1 
Rebound burst 0.03 0.25 52 0.1 
Threshold variability 0.03 0.25 60 2.3 
Bistability I 1.5 60 26.1 0.05 
Bistability II 1.5 60 26.1 0.05 

Notes: Parameters marked with an asterisk indicate parameters that differ from those used in Izhikevich (2004). Additionally, only a single form of bistability (bistability I) was presented in Izhikevich (2004).

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