Multi-scale Models of the Cerebellum: Role of the Adaptive Filter Model

1. Multi-scale Models of the Cerebellum: Role of the Adaptive Filter Model Paul Dean, Christian Rössert & John Porrill University of Sheffield REALNET

2. Adaptive Filter Models of the Cerebellum • First proposed by Fujita in 1982, based on the original Marr-Albus framework • We have argued that many models of the cerebellar role in motor control (especially eye or arm movement) are based in the adaptive filter • Such popularity would suggest that the adaptive-filter model probably has a role to play in multi-scale modelling of the cerebellum • What role? What is the basis for the popularity? Dean, P., Porrill, J., Ekerot, C. F., & Jorntell, H. (2010). The cerebellar microcircuit as an adaptive filter: experimental and computational evidence. Nature Reviews Neuroscience, 11(1), 30-43. Sardinia 2013

3. Fixed Filter • Fixed filters are familiar from audio • Example here (B, C) a low-pass filter that selectively attenuates high frequencies • (Can also be equivalently described in terms of its impulse response (A)) Sardinia 2013

4. Adjustable Filters • Again familiar from audio • Knobs (A) to alter gain (volume) and frequency response (B) Sardinia 2013

5. Adjustable Filters • Can be quite fancy, e.g. a graphic equalizer • Gain of individual frequency channels adjustable for the ultimate listening experience • BUT – still has to be adjusted by hand Sardinia 2013

6. Adaptive Filters • What we want is an adjustable filter where the adjustments are made automatically • Therefore need some sort of ‘adjuster signal’, to tell the filter what to do • Demonstrated by the Analysis-Synthesis filter Sardinia 2013

7. Analysis-Synthesis Filter • Input ‘analysed’ into component signals using a bank of fixed filters (here leaky integrators). • Components are then weighted and recombined (synthesised) to produce the output • Weight values can be adjusted; allows shape of output to be altered. Output is in effect ‘sculpted’ from components. Sardinia 2013

8. How are Weights Adjusted? Adjuster Signal • This is where the adjuster signal comes in • Each weight receives the same adjuster signal (more usually referred to as ‘error’ or ‘teaching’ signal) Sardinia 2013

9. Learning Rule • The adaptive filter changes its weights according to the correlation between input component and teaching signal, i.e. • if positive correlation, reduce weight • if negative correlation, increase weight • Learning stops when there is no correlation between component and teaching signals • In effect, the analysis-synthesis filter implements a decorrelation algorithm Sardinia 2013

10. Adaptive Filters Work • Adaptive filters are very widely used in signal processing – e.g. communications, radar, sonar, navigation, seismology, biomedical engineering, and financial engineering • As already mentioned, widely used to model the cerebellum in the control of eye, eyelid and arm movements • More recently, shown to be a good candidate for the adaptive element in “Internal Models” Sardinia 2013

11. The Forward Model • Example of in internal model – the ‘forward model’ • Suggested in relation to cerebellum by e.g. Miall, Wolpert • Use to predict sensory effects of movement • Useful for e.g. distinguishing external sensory signals from those produced by one’s own movement Miall, R. C., & Wolpert, D. M. (1996). Forward models for physiological motor control. Neural Networks, 9(8), 1265-1279. Sardinia 2013

12. Role of Adaptive Filter • Can show that the adaptive filter is capable of learning forward models • Expands cerebellar role from motor control to sensory prediction (and possibly ‘cognitive’ prediction?) Porrill, J., Dean, P., & Anderson, S. R. (2013). Adaptive filters and internal models: Multilevel description of cerebellar function. Neural Networks, in press. Sardinia 2013

13. How Realistic is the Adaptive Filter Model? • Analysis: granular layer produces components of input (mossy fibre) signals • Components weighted by parallel-fibre Purkinje cell synapses • Weights adjusted by climbing fibre signal • Purkinje cell combines weighted components to produce output. Sardinia 2013

14. Explains Two Striking Features • A very large number of fixed filters are required at the analysis stage to ensure adequate coverage of all contingencies – in biology, the precise nature of the required response cannot be known in advance • Granule cells constitute ~80% of neurons in the human brain (Herculano-Houzel 2010) • The adjuster signal must not interfere with system output, but must be able to affect all weights • Seems to fit climbing fibre properties Herculano-Houzel, S. (2010). Frontiers in Neuroanatomy, 4, Article 12. Sardinia 2013

15. Broad-Brush Realism? • “The structure of the granular layer network and its mossy fibre inputs is well suited for spreading diverse sets of information (referred to here as ‘diversity spreading’).” (p.625) • The huge diversity of parallel fibre codings, which are widely distributed over the molecular layer, has the advantage that guiding signals (provided by climbing fibres) can select and sculpt those codings that are needed to improve behaviour as required in a particular spatiotemporal context.” (p.630). • Consistent with analysis-synthesis adaptive filter Gao, Z. Y., van Beugen, B. J., & De Zeeuw, C. I. (2012). Distributed synergistic plasticity and cerebellar learning. Nature Reviews Neuroscience, 13(9), 619-635. Sardinia 2013

16. What’s the Problem? • Although there may be a broad sense in which adaptive filter models are ‘realistic’, the details are clearly not realistic • Synapses can be either positive or negative • Firing rates not spikes • Neither single neurons nor populations represented explicitly • However, this can’t be simply be solved by switching to current very detailed compartmental models - it can be difficult to determine just what these can do functionally • Problem: • Abstract models can be used for control but lack detail • Detailed models have not been shown to be capable of control Sardinia 2013

17. Bridging the Gap • Natural question: is there a suitable intermediate level of modelling? • Sydney Brenner: “There is a theory called the ‘cell theory’ that is about 150 years old. So I think studying the cell gives the proper perspective. You can then look downwards onto the molecule and upwards to the organism. So it is neither top down nor bottom up, rather it is middle out, and I think that is going to be the correct approach” Sardinia 2013

18. Bridging the Gap • Criterion #1: neurons and synapses represented explicitly but in as simplified a form as possible • Criterion #2: network’s signal processing characteristics can be specified (and shown to be capable – or not – of adaptive filter functionality) Sardinia 2013

19. Possible Benefits • May be possible to assess computational impact of specific additional detailed features • Are there features related to e.g. homeostasis rather than computational power? • Are there features that enable the circuit to do computations that adaptive filters cannot do? Sardinia 2013

20. Examples • Update of a model started in 1991 • Integrate and fire neurons • Used for classical conditioning of eyeblink • As yet no analysis of its computational properties? Sardinia 2013

21. Examples • Conductance-based integrate and fire neurons • Used for eyeblink conditioning and OKR • Talked about this morning Sardinia 2013

22. Examples • Integrate and fire neurons using EDLUT simulator • Used to investigate possible roles of granular-layer plasticity • Will be talked about shortly Sardinia 2013

23. Question • Candidates for bridging the gap between abstract (adaptive-filter)and detailed (compartmental) models • How can they be used to increase understanding of the connection between features at the cellular level, and algorithmic competence? Sardinia 2013