Canard Theory in Computational Neuroscience

Canard phenomenon has recently been suggested in the context of three-dimensional systems of singularly perturbed ordinary differential equations. Canards first introduced in terms of the classical canard phenomenon in the two-dimensional system such as van der Pol oscillator. However, this phenomenon only indicated the transition from a state of small-amplitude oscillatory created in a Hopf bifurcation to a large-amplitude relaxation oscillatory for an extremely small range of a control parameter. Therefore, two-dimensional systems show either small-amplitude oscillations (subthreshold oscillation) or large-amplitude oscillations but no MMOs.  In the three-dimensional systems, a specific type of canard called canards of folded node can be the origin of MMOs. A basic insight about MMOs in this system is that a state of the system is dynamically changed from STO to a large amplitude oscillation so that the system is return from large relaxation oscillation to the basin of attraction of STO state. 

(From "The Effect of Input Current on Canard-Induced Mixed-Mode Oscillation in Layer II Stellate Cell", the 10th Asian Control Conference 2015 (ASCC 2015), Kota Kinabalu, 31st May - 3rd June 2015, IEEE. Babak V.Ghaffari, M.Kouhnavard, T. Kitajima)

Resonator: Calcium channels or Calcium-dependent Potassium Channels

   

From the pharmacological point of view, calcium currents cannot be blocked in PD neurons without removing the calcium-dependent potassium current. Thus, the effects of calcium (ICa) and calcium-dependent potassium (IKCa) currents on resonant properties remain unclear. By taking advantage of Hodgkin-Huxley-type model of neuron and its equivalent RLC circuit, we examine the effects of changing resting membrane potential and those ionic currents on the resonance. Results show that changing the resting membrane potential influences the amplitude and frequency of resonance so that the strength of resonance (Q-value) increases by both depolarization and hyperpolarization of the resting membrane potential. Moreover, hyperpolarization- activated inward current (Ih) and ICa (in association with IKCa) are dominant factors on resonant properties at hyperpolarized and depolarized potentials, respectively. Through mathematical analysis, results indicate that Ih and IKCa affect the resonant properties of PD neurons. However, ICa only has an amplifying effect on the resonance amplitude of these neurons. 

(From "Mathematical Modeling of Subthreshold Resonant Properties in Pyloric Dilator Neurons", BioMed Research International 04/2015; 2015. DOI:10.1155/2015/135787. Babak V.Ghaffari, M.Kouhnavard, T. Aihara, T. Kitajima)