modeling stochasticity and gap junction dynamics: integrate and fire model
TRANSCRIPT
Modeling Stochasticity and Gap Junction Dynamics : Integrate and Fire Neuron Model
Sai Patkar, Guruprasad Krishnamurthy,Dharma Teja Varapula, Bharadwaj Nandakumar
Krishnamoorthy
Model Objective
● Simulate firing of neuron taking into consideration ○ spike rate adaptation○ stochasticity ○ gap junction dynamics
Action Potential
http://www.physiologyweb.com/lecture_notes/neuronal_action_potential/neuronal_action_potential_important_features.html
Figure 1. Neuronal Action Potential
Basic Integrate and Fire Model
● The solution above holds when Ie is independent of time so that integration of (2) is easy
● The above Integration and Fire model is simple enough to simulate a group of neurons
● Doesn’t explain spike-rate adaptation
Spike-Rate Adaptation
Figures show variation of inter-spike interval with time - the inter-spike interval (ISI) in case of no adaptation exhibits a constant value with time (tleft) and the ISI in case of adaptation shows a gradual increase to a constant value after a considerable time period.
Alternative Model 1Leaky Integrate and Fire with spike-rate adaptation (sra) - LIF
● To simulate sra, a compensating current with conductance gsra is added
● This conductance can be modeled as a potassium ion conductance with EK being its resting potential inside the cell
Alternative Model 2Adaptive Exponential Integrate and Fire (AdEx)[2]
● Describes a more realistic action potential due to exponential● Describes spike trains with 96 % accuracy[2]
● LIF model can be derived by making ΔT → 0● sra is accounted for in a similar way as the LIF model with a
constant gL
Proposed Model - GAP jn.
Current across GAP junction can be modeled as
We assume no resistance between the synapses and somaof both the cells, and represent the current, i, as Ie.Hence, (Rs is the resistance offered by the synapse)
The model then becomes:
Vpre Vposti
© 2011 Pearson Education
Proposed Model - Stochasticity● Noise is introduced as Poisson (G) and Gaussian (ξ) processes as
follows (Stein’s Model)[15]:
● µ is the input to which the noise is added● In this case, µ is the input current Ie● The model so obtained will not be deterministic unlike the earlier
models thus taking into account the stochastic nature of neural spiking
Results and Discussion
The action potential (spike) due to a simulated input current from a GAP junction using the developed model.
Results and Discussion
The figure on the left side represents the deterministic output from the AdEx model and the figure on the right side represents the stochasticity-included output of our model.
Results and Discussion
The ISI distributions for a non-stochastic model (AdEx) on the left side and the ISI distribution for the stochastic model (Our model). It can be noticed that with the addition of Poission-based noise, the ISI distribution follows Gamma distribution
Results and DiscussionStochasticity-included output (below) vs the experimental data (top).
Results and DiscussionThe results from the proposed model: The above panels show the different types of spike trains that can be obtained with the model by choosing appropriate set of parameters. The parameters have been chosen as described by Naud et al[7].
Sensitivity Analysis
Sensitivity analysis done for three parameters, gL, Cm, a
Possible Improvements● The input current from a synapse can be better modeled
by considering the passive and active conductances between soma and synapse
● The proposed single neuron model can be extended to a group of neurons to be more beneficial computationally
● Stochasticity can be introduced to model refractoriness
References1: Coombes, S., Zachariou, M., M. Zachariou. Gap Junctions and Emergent Rhythms: Coherent Behavior in Neuronal Networks. Springer Series in Computational Neuroscience. 2009. Vol. 3, pg. 77-94.2: Brette R. and Gerstner W. (2005), Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity, J. Neurophysiol. 94: 3637 - 3642.3: Dayan, P. & Abott, L. F. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. The MIT Press. 2001.4: Gerstner, W. and Kistler, W.M. (2002). Spiking neuron models: single neurons, populations, plasticity, Cambridge University Press, Cambridge5: "Important Features of the Neuronal Action Potential - Neuronal Action Potential - PhysiologyWeb." Important Features of the Neuronal Action Potential - Neuronal Action Potential - PhysiologyWeb. Physiology Web, 5 July 2012. Web. 10 Mar. 2014.<http://www.physiologyweb.com/lecture_notes/neuronal_action_potential/neuronal_action_potential_important_features.html>.6: Burkitt, A. N. "A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input." Biological Cybernetics 19th ser. 95.1 (2006): 1-8. 19 Apr. 2006. Web. 7 Mar. 2014.7: Naud, R., Marcille, N., Clopath, C., Gerstner, W. Firing patterns in the adaptive exponential Integrate and Fire model. Biological Cybernetics. 2008. Vol. 99, pg. 335-347.8: Fourcaud-Trocme N., Hansel D., van Vreeswijk C., and Brunel N. (2003), How spike generation mechanisms determine the neuronal response to fluctuating inputs, J.Neuroscience 23:11628-11640.9: Izhikevich, E.M. (2001), Resonate-and-fire neurons, Neural Networks, 14:883-894.10: Freeman, S. Biological Science. Pearson Prentice Hall, 2nd edition. 2005.11: Hormuzdi SG, Filippov MA, Mitropoulou G, Monyer H, Bruzzone R. Electrical synapses: a dynamic signaling system that shapes the activity of neuronal networks. Biochim. Biophys. Acta. 2005. 1662 (1-2): 113–37.12: M. W. Oram , M. C. Wiener , R. Lestienne , B. J. Richmond, Stochastic Nature of Precisely Timed Spike Patterns in Visual System Neuronal Responses ,Journal of Neurophysiology Published 1 June,1999Vol. 81no. 3021-3033, http://dx.doi.org/10.1038/nrn306113: Mark D. McDonnell & Lawrence M. Ward, The benefits of noise in neural systems: bridging theory and experiment, Nat Rev Neurosc, 2011/07//print14: Chapeau-Blondeau et.al, Stochastic resonance in a neuron model that transmits spike trains, PhysRevE.53.1273, American Physical Society, 10.1103/PhysRevE.53.1273 15: V. Di Maio, P. Lánský and R. Rodriguez, Different Types of Noise in Leaky Integrate-and-Fire Model of Neuronal Dynamics with Discrete Periodical Input, Gen. Physiol. Biophys. (2004), 23, 21|38 16: E. De Schutter, J.M. Bower, An Active Membrane Model of the Cerebellar Purkinje Cell : II. Simulation of Synaptic Responses, JOURNAL OF NEUROPHYSIOLOGY Vol. 71, No. 1, 401-419 January 1994, The American Physiological Society, 1994.