Basal Ganglia. Input: Cortex Output: Thalamus motor and sensory gajng. Striatum Caudate Nucleus Putamen Globus Pallidus

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1 An Interac*ve Channel Model of the Basal Ganglia: Bifurca*on Analysis Under Healthy and Parkinsonian Condi*ons Robert Merrison- Hort, Nada Yousif, Felix Njap, Ulrich G. Hofmann, Oleksandr Burylko, Roman Borisyuk Michael Meyers

2 Basal Ganglia Input: Cortex Output: Thalamus motor and sensory gajng Striatum Caudate Nucleus Putamen Globus Pallidus Interna (GPi) Externa (GPe) Subthalamic nucleus (STN) SubstanJa Nigra Pars compacta (SNc) Pars rejculata (SNr) hpp://en.wikipedia.org/wiki/file:basal_ganglia_circuits.svg

3 Parkinson s Disease Symptoms: bradykinesia and rigidity Death of SNc neurons β- band synchrony increased in STN and GPi local field potenjal (LFP) 8-30 Hz Correlates with symptom severity Product of local neural acjvity? Causes symptoms of Parkinson s? This paper looks for the source GPe- STN coupling can mathemajcally produce β- band oscillajons Requires STN self- excitajon No experimental evidence of this

4 β- band Synchrony How to resolve need for STN self- excitajon without gap juncjons or local axon collaterals? RelaJvely isolated channels in basal ganglia STN excites GPe, which inhibits neighboring channel GPe, releasing neighboring STN from inhibijon (rebound firing)

5 Model τ s x i = x i + Z s ( w ss x i w gs y i +I) τ g y i = y i + Z g ( w gg y i + w sg x i α w gg j L i y j ), i=1,2, N Z j (x)= 1/1+exp ( a j (x θ j )) 1/1+ exp ( a j θ j ), j {s,g} Line connecjvity Circle ConnecJvity L i ={ {i 1,i+1}, 1<i<N {i+1}, i=1 {i 1}, L i ={ {i 1,i+1}, i=n 1<i<N {i+1,n}, i=1 {i 1,1}, i=n

6 Model Parameters Healthy Parkinsonian Both w gg w gs w sg τ s ms τ g ms a s θ s a g θ g - - 2

7 Analyses First in an isolated channel Without STN self- excitajon With STN self- excitajon and healthy parameters With STN self- excitajon and Parkinsonian parameters Frequency analysis Coupled channels (no STN self- excitajon) Healthy Parkinsonian

8 Isolated Channel w/o Self Excita*on Analysis τ s x i = x i + Z s ( w ss x i w gs y i +I) τ g y i = y i + Z g ( w gg y i + w sg x i α w gg j L i y j ), i=1,2, N Set α=0 Set w ss =0 [ 1/ τ s & a s w gs e a s ( θ s I+ w gs y i ) / τ s ( e a s ( θ s + w gs y i I) +1) a g w sg e a g ( θ g NegaJve trace PosiJve determinant

9 Isolated Channel w/o Self Excita*on Analysis τ s x i = x i + Z s ( w ss x i w gs y i +I) τ g y i = y i + Z g ( w gg y i + w sg x i α w gg j L i y j ), i=1,2, N Authors don t exclude possibility of pairs of stable/unstable limit cycles

10 Healthy, Isolated Channel w/ Self Excita*on τ s x i = x i + Z s ( w ss x i w gs y i +I) τ g y i = y i + Z g ( w gg y i + w sg x i α w gg j L i y j ), i=1,2, N w ss is now a bifurcajon parameter

11 Healthy, Isolated Channel w/ Self Excita*on

12 Healthy, Isolated Channel w/ Self Excita*on

13 Healthy, Isolated Channel w/ Self Excita*on 1. w ss <2.191 Fix I = 1 and look at the five regions

14 Healthy, Isolated Channel w/ Self Excita*on Fix I = 1 and look at the five regions 1. w ss < <w ss <2.753

15 Healthy, Isolated Channel w/ Self Excita*on Fix I = 1 and look at the five regions 1. w ss < <w ss < <w ss <2.783

16 Healthy, Isolated Channel w/ Self Excita*on Fix I = 1 and look at the five regions 1. w ss < <w ss < <w ss < <w ss <2.822

17 Healthy, Isolated Channel w/ Self Excita*on Fix I = 1 and look at the five regions 1. w ss < <w ss < <w ss < <w ss < <w ss

18 Healthy, Isolated Channel w/ Self Excita*on SJll no stable oscillajons Fix I = 1 and look at the five regions 1. w ss < <w ss < <w ss < <w ss < <w ss

19 Parkinsonian Isolated Channel

20 BifurcaJons w/ changing w ss when I=3.5 Parkinsonian Isolated Channel SNIC at I=3.5 w ss =9.705

21 Parkinsonian Isolated Channel Stable Oscilla*ons Region D Region C

22 Parkinsonian Isolated Channel Stable Oscilla*ons Frequency Analysis Simulate system at points on a mesh in parameter space FFT used to generate power spectrum Frequency and Amplitude of highest power oscillajon visualised

23 Parkinsonian Isolated Channel Stable Oscilla*ons Frequency Analysis

24 Coupled Channel Frequency Analysis Repeated same frequency analysis for 5 coupled channels in line topology Set w ss =0 in line with biological observajons Replace with α Took too long to reproduce τ s x i = x i + Z s ( w ss x i w gs y i +I) τ g y i = y i + Z g ( w gg y i + w sg x i α w gg j L i y j ), i=1,2, N

25 Summary Healthy STN- GPe network cannot oscillate in isolated channel condijon I found oscillajons in this condijon, but they are unstable Parkinsonian STN- GPe network can oscillate with isolated channels oscillajons are stable and in the beta band but this depends on self- sjmulajon by STN (which does not occur physiologically) Networked STN- GPe channels don t require self- sjmulajon oscillajons sjll in beta band for healthy condijons much larger oscillatory region in parkinsonian condijon much of parkinsonian oscillajon is faster than beta band More connected networks of channels may slow down oscillajons into the beta band In Parkinson s, it is possible for beta band oscillajons to originate from GPe- STN interacjon