Structure of Correlations in Neuronal Networks

Transcription

1 Structure of Correlations in Neuronal Networks Krešimir Josić University of Houston James Trousdale (UH), Yu Hu (UW) Eric Shea-Brown (UW)

2 The Connectome Van J. Wedeen, MGH/Harvard

3 The Connectome Van J. Wedeen, MGH/Harvard Song, et al. 2004

4 Recordings from neurons

5 Recordings from neurons

6 Recordings from neurons

7 Recordings from neurons Are neuronal responses dependent or independent? Tolias, Dragoi, Smirnakis, Angelaki,...

8 Recordings from neurons Are neuronal responses dependent or independent? Tolias, Dragoi, Smirnakis, Angelaki,...

9 Recordings from neurons How are structure and dynamics related in neuronal networks? Are neuronal responses dependent or independent? Tolias, Dragoi, Smirnakis, Angelaki,...

10 Recordings from neurons How are structure and dynamics related in neuronal networks? Synchrony - is probably atypical Are neuronal responses dependent or independent? Tolias, Dragoi, Smirnakis, Angelaki,...

11 Correlation - a measure of dependence Spike Count Correlation neuron 1 neuron 2

12 neuron 1 neuron 2

13 neuron 1 neuron 2

14 neuron 1 neuron 2

15 neuron 1 neuron 2

16 Correlations neuron 1 neuron 2 n - (random) number of spikes of neuron i during a time T. i Correlation coefficient of the output is T ρ T = Cov(n 1,n 2 ) Var(n1 )Var(n 2 )

17 Correlations neuron 1 neuron 2 n - (random) number of spikes of neuron i during a time T. i Correlation coefficient of the output is T Spikes fired by cell 2 ρ T = Cov(n 1,n 2 ) Var(n1 )Var(n 2 ) Spikes fired by cell 1 low correlation

18 Correlations neuron 1 neuron 2 n - (random) number of spikes of neuron i during a time T. i Correlation coefficient of the output is T Spikes fired by cell 2 ρ T = Cov(n 1,n 2 ) Var(n1 )Var(n 2 ) Spikes fired by cell 1 high correlation

19 Short vs long timescale correlations

20 t correlation c 0 Cross-Correlation Function 0.3 Input correlation c Spike train A Long-timescale correlation 1 ) 40 t -3 t t 2 3 Spike train B Conditional probability of w do you need to normalize the resulting histogram to get h B,A (t)? spike in B, given spike in A. Short-timescal (synchrony) sed in proving relation Eq. (4) can again be used to relate the crossith the cross-covariance density c A,B (t) =r B h A,B (t) r A r B, where 10 msr A and e rates of the two processes. Here c A,B (T )( t) 2 can again be interpreted covariance of N A (t + T,t + T + t) and N B (t, t + t).

21 t correlation c 0 Cross-Correlation Function 0.3 Input correlation c Spike train A Long-timescale correlation 1 ) 40 t -3 t t 2 3 Spike train B Conditional probability of w do you need to normalize the resulting histogram to get h B,A (t)? spike in B, given spike in A. Short-timescal (synchrony) sed in proving relation Eq. (4) can again be used to relate the crossith the cross-covariance density c A,B (t) =r B h A,B (t) r A r B, where 10 msr A and e rates of the two processes. Here c A,B (T )( t) 2 can again be interpreted covariance of N A (t + T,t + T + t) and N B (t, t + t).

22 ut correlation c ) 40 f i (t) = t 2 cross-correlation t function C -3 ij (τ) is defined as t 3 j (J ij y j )(t), where J ij (t) = S,j 0 The Cross-Correlation N N matrix J contains the synaptic kernels, while the matrix Function,C weights, and henceinput definescorrelation the networkc architecture. i,j (τ) In particular, W ij = of a synaptic connection from cell j to cell i. Table 1 provides an overview of all parameters and variables. Measures of spike time correlation We quantify dependencies Spike train A between the responses of cells in the network u and cross-correlation functions [Gabbiani and Cox, 2010]. For a pair of sp After normalization C ij (τ) =cov(y i (t + τ),y j (t)). The auto-correlation Spike train Bfunction C ii (t) is the cross-correlation between a s C(t) is the matrix of cross-correlation functions. Denoting by N yi (t 1,t 2 ) Conditional of spikes probability over a timeof window [t 1,t 2 ], the spike count correlation, ρ ij (τ), w do you need to normalize is defined theas, resulting histogram to get h B,A (t)? spike in B, given spike in A. cov N yi (t, t + τ),n yj (t, t + τ) sed in proving relation Eq. (4) can again be ρ ij used (τ) = to relate the crossith the cross-covariance density c A,B (t) =r B h A,B (t) r A var r B, where 10 (N yi ms (t, r A t + and τ)) var N yj (t, t + τ) e rates of the two processes. We assume Here stationarity c A,B (T )( t) of 2 the canspiking again processes be interpreted so that ρ ij does not dep covariance of N A (t + covariance T,t + T + is t) related and Nto B the (t, t cross-correlation + t). function by [Bair et al., b] 7 cov N y (t, t + τ),n y (t, t + τ) τ = C ij (s)(τ s

23 Correlations Impact Neural Computation A B J* (binary input) 0.5 h Cov h h J Cov Reliability 0 Cov(h) -0.5 D E Tkačik, et al. 2010

24 Models of Neurons - Integrate and Fire dv dt = V τ m +Ψ(V )+µ + 2Dη(t) V (t) =V θ V (t + )=V reset Subthreshold membrane potential Fire and Reset V θ V reset Rate r - number of spikes per second

25 Linear response kernel, A(t),X(t)

26 Linear response kernel, A(t),X(t)

27 Linear response kernel, A(t),X(t)

28 Linear response kernel, A(t),X(t)

29 Linear response kernel, A(t),X(t) r(t) =r 0 +(A X)(t), in the absence of the si

30 Structure or correlations in networks dv i dt = V i τ m +Ψ(V i )+ µ + 2Dη(t)+(f i f i ) spike generating current synaptic connections y j (t) = i f i (t) = j δ(t t j i ) (J ij y j )(t), output spike train of cell j synaptic coupling J ij (t) = t τ W D,j ij exp t τ D,j τs,j 2 τ S,j t τ D,j 0 t<τ D,j ynaptic kernels, while the matrix W contains the syn Nykamp

31 Can we estimate the correlation structure? The output of a model neuron is a spike train y j (t) = i δ(t t j i ) Linear response gives the output rate as r(t) =r 0 +(A X)(t), in the absence of the si

32 Can we estimate the correlation structure? The output of a model neuron is a spike train y j (t) = i δ(t t j i ) Linear response gives the output rate as r(t) =r 0 +(A X)(t), in the absence of the si How do we use this to compute the cross-correlation? Idea goes back to Lindner, Doiron, Longtin, 2005

33 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) y 0 (t) = i δ(t t 0 i )

34 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) +X(t) y 0 (t) = i δ(t t 0 i )

35 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) +X(t)

36 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) +X(t) y(t) = i δ(t t i )

37 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) +X(t) y(t) = i δ(t t i ) Use linear response to obtain a mixed point/continuous process y(t) y 1 (t) =y 0 (t)+(a X)(t). tion of the spike train generated

38 Can we estimate the correlation structure? dv dt = V τ m +Ψ(V )+µ 0 + 2Dη(t) +X(t) y(t) = i δ(t t i ) Use linear response to obtain a mixed point/continuous process y(t) y 1 (t) =y 0 (t)+(a X)(t). tion of the spike train generated Which averages out to the right thing r(t) r 0 +(A X)(t)

39 Approximate network correlations The linear response approximation now takes the form yi 1 (t) =yi 0 (t)+ ( Ki,j [yj 0 r j ] ) (t) all inputs K i,j =(A i J i,j )(t) We can use this to approximate the cross-covariances C ij (τ) C 1 ij(τ) =E (y 1 i (t + τ) r i )(y 1 j (t) r j ) = δ ij C 0 ii(τ)+(k ij C 0 jj)(τ)+(k ji C0 ii)(τ)+ k (K ik K jk C0 kk )(τ) Ostojic, Brunel, Hakim, 2009, Trousdale, Yu, Shea-Brown, Josić, 2011

40 Impact of non-immediate neighbors We use an iterative construction y n+1 (t) =y 0 (t)+(k [y n r]) (t) n+1 ( ) = y 0 (t)+ K (k) [y 0 r] k=1 (t) Which gives the n-th approximation to the cross-correlation After taking the Fourier transform, and the limit n C (ω) = lim n C n (ω) =(I K(ω)) 1 C 0 (ω)(i K (ω)) 1.

41 Impact of non-immediate neighbors We use an iterative construction y n+1 (t) =y 0 (t)+(k [y n r]) (t) n+1 ( ) = y 0 (t)+ K (k) [y 0 r] k=1 (t) Which gives the n-th approximation to the cross-correlation After taking the Fourier transform, and the limit n C (ω) = lim n C n (ω) =(I K(ω)) 1 C 0 (ω)(i K (ω)) 1. K i,j =(A i J i,j )(t)

42 The iterative construction Rangan 2009 Pernice, Staube, Cardanobile, Rotter 2011 Trousdale, Yu, Shea-Brown, Josić, 2011

43 The iterative construction Rangan 2009 Pernice, Staube, Cardanobile, Rotter 2011 Trousdale, Yu, Shea-Brown, Josić, 2011

44 The iterative construction Rangan 2009 Pernice, Staube, Cardanobile, Rotter 2011 Trousdale, Yu, Shea-Brown, Josić, 2011

45 The approximation works well Cross-correlation between two excitatory cells as we shift from balance to excess inhibition

46 Expansion in terms of paths through the graph C (τ) = lim n n k,l ( ) K (k) C 0 K (l)t (τ) j a 0 a 1 a 2 i a 1 b 1 i b 2 j K ian 1 K an 1 a n 2 K a1 j C 0 jj K ian 1 K an 1a n 2 K a1 a 0 C 0 a0a0 K a 0 b 1 K b m 2 b m 1 K b m 1 j

47 How does local structure determine correlations? Song, et al. 2005

48 How do small motifs impact the correlation structure?!" #" $" %#&'()#*)" " q div = (Wi,kW 0 j,k)/n 0 3 p 2 i,j,k Sporns and Kötter, 2004

49 How do small motifs impact the correlation structure?!" #" $" $"!" #" $"!" #" %#&'()#*)" " " +,*&'()#*)" +-.#*" q div = (Wi,kW 0 j,k)/n 0 3 p 2 i,j,k q con q ch Sporns and Kötter, 2004

50 Mean correlations in structured networks

51 How do small motifs impact the correlation structure?!" #" $" $"!" #" $"!" #" %#&'()#*)" " " +,*&'()#*)" +-.#*"

52 Correlations with homogeneity C (ω) =(I K(ω)) 1 ỹ 0 (ω)ỹ 0 (ω)(i K (ω)) 1 Assuming homogeneity in uncoupled cells, and evaluating at ω = 0 C (0) = C 0 (0)(I ÃW) 1 (I ÃWT ) 1 After expanding and truncating at second order in connection strength, writing ww 0 = W C C 0 I + ÃwW0 + ÃwW0T + Ãw 2 W 0 W 0T + Ãw 2 W Ãw 2 W 0T 2

53 Averagd network correlations C C 0 I + ÃwW0 + ÃwW0T + Ãw 2 W 0 W 0T + Ãw 2 W Ãw 2 W 0T 2 Averaging over the network C C 0 1 N +2Ãwp +3N Ãw 2 p 2 + N Ãw 2 qdiv +2N Ãw 2 qch.

54 Resumming

55 C C 0 = 1 N 1 1 (NÃw)n L T WnL 0 1+ n=1 1 1 (NÃw)m L T Wm L 0T, m=1 Resumming (NÃw)n+m L T Wn,mL 0 n,m=1 Keeping contribution of second order motifs C C 0 = 1 N NÃw qdiv NÃw NÃw 2 2. p qch

56 C C 0 = 1 N 1 1 (NÃw)n L T WnL 0 1+ n=1 1 1 (NÃw)m L T Wm L 0T, m=1 Resumming (NÃw)n+m L T Wn,mL 0 n,m=1 Keeping contribution of second order motifs C C 0 = 1 N NÃw qdiv NÃw NÃw 2 2. p qch

57 Theory extends to EI % & ' ( +, - %."!"#!*#!+#!,#!-#!.# ) * %%" %&"!/# 01# " /!0123!43" !43" " $%&'(#

58 Conclusion - Linear response theory can be used to understand the statistical structure of population activity. - Cross-correlation functions can be understood in terms of contributions from paths through the network. Thus architecture and population activity can be related. - This local theory applies to any network where interactions can be linearized - There is a lot more to do - see Bullmore and Sporns, Nat Neurosci, 2009

59 University of Houston Manisha Bhardwaj Becky Chen Manuel Lopez Robert Rosenbaum (Pitt) James Trousdale University of Washington, Seattle Eric Shea-Brown Yu Hu University of Pittsburgh Brent Doiron