ZHOU Changsong ( 周昌松） Department of Physics Centre for Nonlinear Studies (CNS)

1. Less is More: Cost-efficient dynamical modes in biological neural network ZHOU Changsong (周昌松） Department of Physics Centre for Nonlinear Studies (CNS) Institute of Computational and Theoretical Studies Hong Kong Baptist University Beijing Computational Science Research Center Collaborators: Dr. Yang Dong-Ping (HKBU) Prof. Zhou Haijun (ITP) Prof. Tang Leihan (HKBU, CSRC)

2. Less is More: Cost-efficient dynamical modes in biological neural network ZHOU Changsong (周昌松） Department of Physics Centre for Nonlinear Studies (CNS) Institute of Computational and Theoretical Studies Hong Kong Baptist University Beijing Computational Science Research Center Collaborators: Dr. Yang Dong-Ping (HKBU) Prof. Zhou Haijun (ITP) Prof. Tang Leihan (HKBU, CSRC)

3. Outline • Introduction: features in neural activity • Individual activity: • low-rate firing • irregularity • Weak pairwise correlations • Collective dynamics: • network oscillations • neuronal avalanches • Our study and numerical results: Integrating the dynamical modes by cost-efficiency trade-off • Summary

4. Brain as Dynamical Complex System D.R. Chialvo “Critical Brain Networks” Physica A 340 (2004) • But brain is a functionalComplex System: • Designed by Great Nature through evolution • Subject to multiple constraints • Optimization of multiple functions • Landscapes of evolution? • Some ideas about the constraints Brain Research: Challenge of the Century: 2012, Human Brain Project, European Flagship 2013, Human Brain Mapping Initiative, USA • High degree of freedom • Strong nonlinearity • Complex Connectivity

5. Energy cost of communication in the brain • Energy demands (50—80%) of spiking: • Restoration of ion movements • Propagation of action potentials • Neurotransmitter uptake Human brain: 2% of the body’s weight 20% of resting metabolism Energy constraint: low firing rate (1~5Hz) From internet

6. Salient features in brain dynamics • Individual activity: • low-rate firing • irregularity • Weak pairwise correlations • Collective dynamics: • network oscillations • neuronal avalanches Must be important for information processing

7. Irregularity of neuronal discharge Different trials with identical stimulus recorded from area MT (V5) of an alert monkey Softky, W.R. and C. Koch J Neurosci, 1993. 13, 334-350 Shadlen M N and Newsome W T, J Neurosci, 1998. 18, 3870–3896

8. The origin of irregularity: E-I balanced network E Patch clamp technique E-I balance: equal average amounts of de- and hyperpolarizing membrane currents External I • Properties: • Sparse network connectivity • Stable at low rate with high fluctuation • Display asynchronous irregular dynamics Shu, Y.S., et. al. Nature, 2003. 423, 288 Robert C. F, et. al. Nature, 2007. 450, 425 Van Vreeswijk, C. , Science, 1996. 274, 1724-1726.

9. Rate tracking in E-I balance regime External input • Cost-efficient: Low rate and sensitive rate response • Cost-efficient? • Waste of materials to achieve only linear response? • The capacity: how neurons can be grouped in a diverse way. • What are the effects of neuronal correlations in information reliability? balanced unbalanced Rapid responses to small input changes Van Vreeswijk, C. , Science, 1996. 274, 1724-1726.

10. Effects of correlations on information processing with rate: • Impair the estimation of information conveyed by the firing rates of neural populations • Limit the efficiency of an organism for performing sensory discriminations • a signature of active information processing? Zohary E. et al. Nature, 1994, 370, 140

11. Decorrelation in densely connected and balanced network Correlations between E and I inputs have a decorrelating effect rin (rout): input (output) spike correlation c: input current correlation Renart, A., et al. Science, 2010. 327, 587-590.

12. Near-zero mean correlation between firing of neurons in the rat neocortex in vivo Asynchronous state “activated” (ACT) state: Rapid eye movement (REM) sleep Up-down transitions “inactivated” (InACT) state: Slow-wave sleep recordings collected with silicon microelectrodes in somatosensory and auditory cortices of urethane-anesthetized rats Renart, A., et al. Science, 2010. 327, 587-590.

13. Weak pairwise correlation is crucial for the firing patterns Boltzmann distribution under maximum entropy assumption: Schneidman E, Nature, 2006, 440, 1007

14. Neural Avalanches: Self-Organized Criticality (SOC): Earthquake in the Brain! J. Beggs, D. Plenz, 2003. J. Neurosci. 23 (35), 11167. E. Gireech, D. Plenz, 2008. PNAS, 105, 7576 T. Petemann, et al, D. Plenz, 2009. PNAS, 106, 15921

15. Neuronal avalanches for information processing Dynamic range Information Capacity Variability of Phase Synchrony Maximal variability of phase synchrony in cortical networks with neuronal avalanches Neuronal Avalanches imply maximum dynamic range in Cortical Networks at Criticality Information Capacity and Transmission Are Maximized in Balanced Cortical Networks with Neuronal Avalanches Shew, W.L., H.D. Yang, T. Petermann, R. Roy, and D. Plenz. J Neurosci, 2009. 29, 15595-15600 Yang, H., W.L. Shew, R. Roy, and D. Plenz. J Neurosci, 2012. 32, 1061-1072 Shew, W.L., H. Yang, S. Yu, R. Roy, and D. Plenz. J Neurosci, 2011. 31, 55-63

16. Collective oscillations Hippocampus in vitro Local field potential (LFP) Human Brain Oscillations (EEG/MEG) Functional role of oscillations: • Timing • Prediction • Coordination • Communication strong gamma (30–40 Hz) oscillation of the LFP, together with low (2 Hz) and irregular firing in pyramidal cells. G. Buzsaki & A. Draguhn, Neuronal Oscillations in Cortical Networks, Science, 304 (2004) G. Buzsaki, Rhythms of the Brain (2006) Linkenkaer-Hansen K, J. Neurosci, 2001, 21: 1370 Fisahn A, et. al. Nature394: 186–189, 1998.

17. Oscillations with sparsely firing neurons Inhibitory-inhibitory coupling Excitatory input Rate model: r(t) instantaneous firing rate Fixed point of steady state: Consider small departures r(t) from r0 : Hopf bifurcation: Oscillation frequency is of the order of the inverse of the delay in the synaptic transmission. N. Brunel and X.J. Wang, J. Neurophysiol. (2003). N. Brunel and V. Hakim, Chaos 18, 015113 (2008); Wang, X.J., Physiol Rev, 90, 1195-1268 (2010).

18. Parallel Lines of Theoretical Research • Sustained activity • & Oscillations • E-I network/balanced state • Asynchronous state • (no correlation, • no avalanches) • Emergence of oscillations • Due to I-I synchronization • (e.g. gamma wave) Van Vreeswijk, C. , Science, 1996. 274, 1724-1726. Brunel, N. and X.J. Wang, J Neurophysiol, 2003. 90, 415-430

19. Parallel Lines of Theoretical Research • Avalanches • & SOC • Branching model • (classical critical state, • no correlation in avalanches) • E-network+plasticity (STD) • (but with unrealistic assumptions) • Inhibitory neurons • seldom considered A. Levina, J. M. Herrmann, and T. Giesel, “Dynamical synapses causing self-organized criticality in neural networks”, Nature Physics 3, 857-860 (2007). S. J. Wang and C. S. Zhou,“Hierarchical modular structure enhances the robustness of self-organized criticality in neural networks”, New J. Phys. 14, 023005 (2012).

20. Understanding Complexity in the Brain • Structure-dynamics-function relationship • Multiple scale • Different features explained by different models • Physical principle vs. biological realism Local circuits Neurons Columns Blue Brain Project: Detailed simulation of a cortical column Human Brain Project Biology: Realization and Implementation of the Principles for Functions Physics: General principles Constrains

21. Questions • Why does neural system generate these dynamic modes altogether? • How are they related to each other? • Can a generic neural network account for these dynamical modes?

22. Low cost Efficient functions Working Hypothesis: a trade-off between Low cost Neurobiological & Dynamical Mechanisms of Structure-Dynamics-Function Relationship Metabolism /Activity Sparse firing Exci. /inhi. neurons Synaptic dynamics (STDP, STD) Structure Economical wiring Small-world/hierarchical /modular ? Information Processing Fast/sensitive response Coherent firing/propagation Synchronization Efficient functions • Can we integrate the dynamical modes in terms of cost-efficiency trade-off? • What are the neurobiological foundation and dynamical mechanism for the cost-efficient dynamical modes? • How are they actually used in information processing?

23. Let us consider generic local neuronal circuits with a balance of excitation and inhibition • Global Network • ~ 1011 neurons • ~ 104-105 synapses per neuron • Generic Local Neural Circuits: • Dense connectivity (P~0.2) • Excitatory: Inhibitory~ 4:1

24. Our framework Energy cost constraint: Low firing rate (1~5 Hz) Information processing: Maximal entropy Irregular Neuronal avalanches Collective oscillations Cost-efficiency? Underlying mechanisms? Basic characteristics of generic neuronal networks Excitation-inhibition (E-I) balance Realistic synaptic dynamics

25. Conductance-based neuronal network model Random E-I recurrent network Fast excitatory synaptic current (2~5 ms) Time course of synaptic conductance: Slow inhibitory synaptic current (5~10ms) • N=1000 neurons (80% exc) • Random network • Connectivity p=0.2 • Poisson input Brunel, N. and X.J. Wang, J Neurophysiol, 2003. 90, 415-430

26. Dynamical Equations

27. Result I: E-I balanced state maintains low-rate firing and irregular individual activity (weak background input fex=20Hz)

28. Result II: Irregular firing and weak synchronization Synchrony: probability of spike alignment CV=standard deviation/average of inter-spike interval (ISI) Irregularity: CV>=1

29. Result III: Neuronal avalanches & Oscillations • Coexistence of • Critical avalanche • Oscillation (alpha wave) • Lowest firing rate • In biological parameter region

30. Result IV:Cost-efficiency trade-off Distance Efficiency=Npatt x Hpatt/Nspikes Npatt:Number of non-trivial patterns Hpatt: Entropy of patterns Nspike: Total number of spikes used

31. Result VI: Stimulated activity • Coexistence of • Critical avalanche • Oscillation (Beta/Gamma wave) • Lowest firing rate • In biological parameter region

32. Cost-efficiency trade-off with stimulation

33. Result V: Robustness of cost-efficient dynamics Mixed 2D parameter space:

34. Cost-efficiency trade-off Cost-efficient dynamical states can be self-organized by E-I balance with suitable synaptic time constants

35. Result V: Robustness of cost-efficient dynamics at High external inputs (fex=200Hz) With external inputs, the region for coexistence is shifted

36. Cost-efficiency trade-off

37. Preliminary understanding Correlated input enhances neuron response rate A. Kuhn, S. Rotter, A. Aertsen, Neurocomputing, 2002 With suitable weak correlation, a network can sustain at lowest rate

38. Challenge: How to treat firing rate and correlation (dynamical clustering) self-consistently in the biological recurrent networks

39. Unification of dynamical features by cost-efficiency trade-off • A basic neural circuit can simultaneously account for salient features in neural activity (qualitatively and quantitatively) • Capturing the leading factors • The unification of the dynamical modes to achieve cost-efficiency trade-off • Challenging for complete theoretical analysis Sparsely synchronous oscillations Asynchronous state critical branching Synaptic plasticity Theory? Theory?

40. Summary • Cost-efficiency trade-off plays crucial role in neural activity • A framework and tool to better understand neural activities • Challenges: • Quantify cost and efficiency properly: concrete information processing • Theoretical analysis (on going with Haijun and Leihan) • Information processing using highly fluctuating dynamics (learning, memory, plasticity)

41. Thank you!

42. Future Roadmap: bottom-up linking models with data 7. fMRI data: functional connectivity, energy scaling 5. Scaling of activity: Gray matter-white matter scaling 4. Cost: constructive integration of wiring and metabolic costs 8. EEG data: ongoing activity vs. stimulus induced activation 6. Multimodal/Multiscale Activity: EEG vs. fMRI 3. Human Brain Connectivity: Multiple constraints and wiring economics Large-scale network Collaboration: High-performance computing Collaboration: Multimodal analysis of imaging data/disease Real connectivity Collaboration: Nonlinear Dynamics / Statistical Mechanics/ Multi-scale/Coarse-graining Analysis E-I network (Building Blocks) • Information • processing (plasticity): • Stability of memory vs. flexibility of learning 2. Normal vs. abnormal states: Altered dynamics, memory and learning

43. Theoretical Analysis • Asynchronous state in E-I balanced neural networks • Sparsely network • Dense network • Correlation and Synchrony between neural spike trains

44. Asynchronous state in E-I balanced neural networks

45. Asynchronous state in sparse E-I balanced neural network Instantaneous fields: Binary state neuron: Mean field equation: Sparseness assumption (K<<N): E K K External K I Balanced steady state: Van Vreeswijk, C. and H. Sompolinsky, Science, 1996. 274, 1724-1726

46. Asynchronous state in dense E-I balanced neural network (K=0.2N) Network state probability: Master equation: Average activity and joint probability: Renart, A., et al. Science, 2010. 327, 587-590.

47. Mean field equations: Shared inputs Neuronal correlations Asynchronous state assumption: Stationary state: neurons are effectively independent central limit approximation

48. Correlation and Synchrony between neural spike trains:two neurons driven by shared input

49. Integrate-and-Fire (IF) Neurons IF neuron model: Spike count correlation: Cross spectrum: Wiener-Khinchine Theorem: J. de la Rocha, et al. Nature, 2007. 448, 802. E. Shea-Brown, et al. PRL, 2008, 100, 108102

50. Threshold Neuron Models Spike correlations: Conditional firing rate Leaky integrator and spike train: Weak correlations: Current and voltage correlations: Gaussian ensemble: T. Tchumatchenko, et al.PRL, 2010, 104, 058102