Dynamic Modeling of Brain Activity

Transcription

1 0a Dynamic Modeling of Brain Activity EIN IIN PC Thomas R. Knösche, Leipzig

2 Generative Models for M/EEG

3 4a Generative Models for M/EEG states x (e.g. dipole strengths) parameters parameters (source positions, conductivities etc.) Input u f(x,,u) =0 y = g(x, ) Stimulation Sources with constraints Observer Measurement

4 4b Generative Models for M/EEG states x (e.g. dipole strengths) parameters parameters (source positions, conductivities etc.) Input u x(t)/ t = f(x(t),u(t), ) y = g(x(t), ) Stimulation Dynamics Observer Measurement

5 4c Generative Dynamic Models Input u States x (e.g. membrane currents and potentials) Parameters (e.g. synapt. parameters etc.) x(t)/ t = f(x(t),u(t), ) Model of the behaviour of nerve tissue over time. Level of description needs to fit spatial and temporal resolution of data. Model parameters should be interpretable (i.e. biologically plausible).

6 Mathematical Formulation of a Neural Mass Model

7 Modelling a Single Neuron impulse rate m(t) weighted summation v( t) m( t) h( t) impulse rate m(t) H t exp( t h( t) 0 ) t t 0 0 m(t) v(t) v(t) m(t)

8 10 Neural Masses Large number of neurons (typically 10 2 to 10 4 ), which... are similar in location, physiology, morphology and connectivity, and can therefore be lumped together and treated as one neuron. states: mean membrane potentials and impulse rates parameters: lumped or equivalent parameters

9 11 Modelling Neural Masses mean impulse rate m(t) mean impulse rate m(t) v( t) m( t) h( t) H t exp( t ) h( t) 0 t t 0 0 m(t) v(t) v(t) m(t)

10 12 Jansen+Rit Model of a Brain Area Pyramidal cells Input from other areas (cortex + thalamus) Excitat.Interneurons (Spiny stellates) Inhibit. interneurons Synapt. time constants : i, e Synaptic strengths : H i, H e Number of synapses : k Output to other areas Freeman, Mass action in the nervous system Lopes da Silva et al Jansen & Rit 1995

11 18a Coupling between Areas - Delays EIN IIN EIN IIN PC PC Short delays (<2 ms) (modeled a dendritic time constant) Long delays (10-30 ms)

12 19 Summary Neural Mass Models Neural masses summarize neurons, which are similar with respect to physiology, morphology and function. The description is analogous to single neurons. Neurons and neural masses are described by rate-to-potential operator (ODE 2. order) and potential-to-rate operator (non-linear function). Brain are is described by standard circuit with pyramidal cells as well as excitatory and inhibitory interneurons. One model can comprise several interconnected areas. Neural mass models describe the dynamic interactions between neural populations and brain areas as system of non-linear ODEs with delays. Neural mass models form a mesoscopic description and therefore are suitable as generative models for EEG and MEG. As observer function classical EEG/MEG forward models are used. Neural mass models are simple (few parameters) and at the same time biologically plausible, i.e., their parameters have a physiological meaning.

13 20 Limitations and Alternatives Mechanisms at microscopic level can not be modeled. Alternative: different data (e.g., single cell recordings) Activity just in some spatially contrained brain areas, hence mechanisms of spatial organisatiion cannot be modeled. Alternative: description by spatio-temporal PDEs or integrodifferential equations neural field models Spike time dependent effects (z.b. spike time dependent plasticity) cannot be accounted for. Alternative: Spiking neuron models, such as integrate-and-fire or Hodgkin-Huxley. Stochastic effects are not modeled. Alternative: population density models (e.g., with Fokker-Planck- Gleichung),

14 Inversion of Neural Mass Models Dynamic Causal Modelling

15 21 We have: Parameter Estimation EEG and/or MEG data y(t) a generatives model for these data m:y(t)=f(x(t),, ) with states x(t) and parameters and additional knowledge on model parameters (priors) We search: estimation of model parameters s and s

16 22 But: Parameter Estimation The generative model cannot predict the data excactly, because the model is a simplification, i.e. it contains inaccuracies.... the data are contaminated by noise. Die priors are specfied in terms of probability distributions. Questions: What is the probability of a candidate solution? What is the most likely solution and how sure can we be about this solution?

17 23g 23e For a model m:y=f(x, ): Likelihood Probability density of the measurements conditional of the model parameters (forward solution) In DCM: Gauss distributed p( y, m) The law of Bayes p( y p( y, m), m) Prior A priori probability density of the parameters In DCM: Gauss distributed p( p( m) m) d Posterior A posteriori probability density of the parameters conditional of the data. In DCM: normalverteilt Model evidence Probability of the data conditional of the model (=p(y m)). = const.

18 28 Solution with Expectation Maximization Algorithm Iterative procedure Determines expectation value and covariance of posterior distribution of parameters Simultaneously estimates parameter distribution and noise covariance. Friston, 2003

19 26 How does DCM Explain Differences between Experimental Conditions? Intrinsic parameters Synaptic strengths and time constants Intrinsic connectivities IFG Extrinsic parameters Extrinsic connectivities Extrinsic transmission delays Input parameters strengths, time constants Modulator (stimulus) Modulator (context) STG A1 STG A1 Observer parameters Dipole positions and directions David et al Driver (stimulus) Input

20 29d Model Selection Using Model Evidence Everything depends on the choice of the model! Which model is the best? p( y, m) p( y, m) p( p( y m) m) Model evidence p ( y m) p( y, m) p( m) d log(p(y m)) Accuracy(m) Complexity(m) Approximation by, e.g.: free energy Akaike criterion Penney et al. 2004

21 30b Model comparison using Bayes factors B i, j p( y p( y m m i j ) ) Kass and Raftery (1993, 1995) Penney et al. 2004

22 31 Summary Dynamic Causal Modelling Bayesian estimation enables the combination of several sources of information (data, prior knowledge). Bayesian estimation considers the uncertainty in data (noise) and prior knowledge and generates a probability density function of possible solutions. DCM assumes that both prior knowledge and noise, and hence also the solution, are normally distributed. Differences in experimental conditions are currently explained through differences in extrinsic connectivity values (modulator) and input strength (driver). The solution is found using the expectation maximization algorithm, which estimates noise covariance and system parameter distributions at the same time. Alternative models are compared using Bayes factors, based on model evidence estimates.

23 What is this good for? One example

24 34 Application I: Schizophrenia Perception Internal concept formation Control Decision Bottom-up processing Top-down processing Schizophrenia (1 % of population) Sensoric input Dima 2009

25 35 Application I: Schizophrenia Hollow mask illusion Dima 2009 Healthy: 23 % Patients: 92 % Recognize mask as hollow

26 36 Application I: Schizophrenia Hypothesis: In healthy subjects strong top-down influence of internal concepts causes the illusion of a true convex face. In schizophrenics this influence is mainly lacking. Modell2 Model 1 SMG Stim V1 LOC IFG IPS Group comparison modulation M1: Controls>Patients (p<0.05) M2: equal Model comparison Controls: M1>M2 (BF = 34) Patients: M2>>M1 (BF = 347) Dima 2009

27 37 Possible Clinical Application Fields Diagnosis tool for psychiatric and neurological diseases. Many diseases coincide with changes in connectivity and synaptic plasticity, e.g. multiple sclerosis, schizophrenia or Alzheimers disease. These disorders are often difficult to diagnose and classify on the basis of their symptoms. Effective mechanisms of pharmacological substances Many psychopharmacological drugs influence the effects of neurotransmitters and hence the connection strengths between neural populations. In order to understand the working mechanisms, biologically realistic models of connectivity are needed.

28 Modeling a Controversal Issue? Modeling is a complicated and risky, yet inevitable, business: Without modeling, no insight is possible. All researchers model, even if they do not use explicit mathematical formulations. With modeling, one can obtain wrong results, if the model assumptions are not true. The result of a model, that means of all scientific research, has to be formulated like under the condition that,

29 Modeling a Controversal Issue? One principal problem with DCM is the model selection: the space of possible models explodes very quickly. Since an incorrect model yields incorrect conclusions without warning, DCM should not be applied in an explorative way. Pros and cons: see Lohmann et al. Critical comments on dynamic causal modelling. (NeuroImage 2012) Friston et al.: Model selection and gobbledygook: response to Lohmann et al. (NeuroImage 2013) Breakspear: Dynamic and stochastic models of neuroimaging data: a comment on Lohmann et al. (NeuroImage 2013)

30 38a References Freeman: Mass Action in the Nervous System. Academic Press 1975 Bible of modeling EEG and mass action in the brain Jansen/Rit: Biological Cybernetics 73, (1995) Introduction of 3-mass models for cortical area. David/Friston: NeuroImage 20, (2003) Neural mass models, coupled brain areas. Spiegler/Kiebel/Atay/Knösche: NeuroImage, in press (2010) Comprehensive analysis of dynamics of Jansen/Rit model Birbaumer/Schmidt: Biologische Psychologie. Springer 1991 Basics of brain anatomix and cell physiology.

31 38b References Marrairos/Stephan/Friston: Didactic review, also covers DCM for fmri Friston: Dynamic Causal Models. In Frackowiak et al., Human Brain Function. Academic Press, 2003 Summary DCM, with EM algorithm. David/Kiebel/Harrison/Mattout/Kilner/Friston: NeuroImage 30, (2006) DCM for EEG and MEG Penney/Stephan/Mechelli/Friston: NeuroImage 22, (2004) Model comparison and selection in DCM Further DCM references: Application example:

32 Thanks for the attention