![Alternative Model 2
Adaptive 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](https://image.slidesharecdn.com/plcuwlznqq6jckiez8k0-signature-755dc48a7d401d2cf8f2a46c5bb7a488da74370747eab3089ac225a666571ca3-poli-150406151108-conversion-gate01/85/Modeling-Stochasticity-and-Gap-Junction-Dynamics-Integrate-and-Fire-Model-7-320.jpg)
![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](https://image.slidesharecdn.com/plcuwlznqq6jckiez8k0-signature-755dc48a7d401d2cf8f2a46c5bb7a488da74370747eab3089ac225a666571ca3-poli-150406151108-conversion-gate01/85/Modeling-Stochasticity-and-Gap-Junction-Dynamics-Integrate-and-Fire-Model-9-320.jpg)
![Results and Discussion
The 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]
.](https://image.slidesharecdn.com/plcuwlznqq6jckiez8k0-signature-755dc48a7d401d2cf8f2a46c5bb7a488da74370747eab3089ac225a666571ca3-poli-150406151108-conversion-gate01/85/Modeling-Stochasticity-and-Gap-Junction-Dynamics-Integrate-and-Fire-Model-14-320.jpg)
Modeling Stochasticity and Gap Junction Dynamics: Integrate and Fire Model
- 1. Modeling Stochasticity and Gap Junction Dynamics : Integrate and Fire Neuron Model Sai Patkar, Guruprasad Krishnamurthy, Dharma Teja Varapula, Bharadwaj Nandakumar Krishnamoorthy
- 2. Model Objective ● Simulate firing of neuron taking into consideration ○ spike rate adaptation ○ stochasticity ○ gap junction dynamics
- 3. Action Potential http://www.physiologyweb. com/lecture_notes/neuronal_action_potential/neuronal_action_potential_important_features.html Figure 1. Neuronal Action Potential
- 4. 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
- 5. 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.
- 6. Alternative Model 1 Leaky 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
- 7. Alternative Model 2 Adaptive 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
- 8. Proposed Model - GAP jn. Current across GAP junction can be modeled as We assume no resistance between the synapses and soma of both the cells, and represent the current, i, as Ie . Hence, (Rs is the resistance offered by the synapse) The model then becomes: Vpre Vpost i © 2011 Pearson Education
- 9. 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
- 10. Results and Discussion The action potential (spike) due to a simulated input current from a GAP junction using the developed model.
- 11. 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.
- 12. 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
- 13. Results and Discussion Stochasticity-included output (below) vs the experimental data (top).
- 14. Results and Discussion The 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] .
- 15. Sensitivity Analysis Sensitivity analysis done for three parameters, gL , Cm , a
- 16. 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
- 17. References 1: 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, Cambridge 5: "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/nrn3061 13: Mark D. McDonnell & Lawrence M. Ward, The benefits of noise in neural systems: bridging theory and experiment, Nat Rev Neurosc, 2011/07//print 14: 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.