Monosynaptic Scalable Architecture Revealed by Transsynaptic Rabies Tracing

Daniel Fürth
Meletis Lab

Network for Networks
12 Feb 2016

daniel.furth@ki.se

  • Yang Xuan
  • Ourania Tzortzi‎
  • Iakovos Lazardis
  • Konstantinos Meletis

 

  • DMC lab

Acknowledgement

Reconstructing brain from sectioned tissue

Tracing the network

Tracing the network

Tracing the network

Scaling rules for rodent brians

Isometric

Allometric

V \propto M^1
V∝M1V \propto M^1
V \propto M^k, \quad k \neq 1
V∝Mk,k≠1V \propto M^k, \quad k \neq 1

Cellular scaling rules for rodent brians

Herculano-Houzel et al. PNAS

Cellular scaling rules

Stevens (2000) Nature

Cellular scaling rules

Stevens (2000) Nature

Cellular scaling rules

Hubel & Wiesel (1974), Blasdel (1986)

Cellular scaling rules

n
nn
n^{1/2}
n1/2n^{1/2}
n^{3/2}
n3/2n^{3/2}

Cellular scaling rules

Stevens (2000) Nature

n \rightarrow n^{1/2} \rightarrow n^{3/2}
n→n1/2→n3/2n \rightarrow n^{1/2} \rightarrow n^{3/2}

Cellular scaling rules

Monosynaptic scaling in mice

k = 1.44, \quad (95\% \text{C.I:} \quad 1.27 - 1.60)
k=1.44,(95%C.I:1.27−1.60)k = 1.44, \quad (95\% \text{C.I:} \quad 1.27 - 1.60)

Synapse density and dendritic branching

Cuntz, Mathy & Häusser (2012) PNAS

Synapse density and dendritic branching

Livneh, Feinstein, Klein and Mizrahi (2009) JoN

Synapse density and dendritic branching

Cuntz, Mathy & Häusser (2012) PNAS

Synapse density and dendritic branching

Cuntz, Mathy & Häusser (2012) PNAS

L \propto bp \propto puncta
L∝bp∝punctaL \propto bp \propto puncta

Self-organizing branching process

p(t+1) = p(t) + \frac{1-\sigma_i(p,t)}{N}
p(t+1)=p(t)+1−σi(p,t)Np(t+1) = p(t) + \frac{1-\sigma_i(p,t)}{N}
P_i(s,p) = s^{-3/2} \sqrt{\frac{2(1-p)}{\pi p}} \exp\left( \frac{-s\ln(4p(1-p))}{-2} \right)
Pi(s,p)=s−3/22(1−p)πpexp(−sln(4p(1−p))−2)P_i(s,p) = s^{-3/2} \sqrt{\frac{2(1-p)}{\pi p}} \exp\left( \frac{-s\ln(4p(1-p))}{-2} \right)
P_i(s) \propto s^{-3/2}
Pi(s)∝s−3/2P_i(s) \propto s^{-3/2}

Self-organizing branching process

Amaral et al. (2000); White, Southgate, Thompson & Brenner (1986)

Mean-field results

Y = \sum\limits_{i;j;k} w_k F_{ij}^k,
Y=∑i;j;kwkFijk,Y = \sum\limits_{i;j;k} w_k F_{ij}^k,

Total output for the network:

I_{ik} = \sum\limits_{j} F_{ij}^k,
Iik=∑jFijk,I_{ik} = \sum\limits_{j} F_{ij}^k,

Total connection of type k for ith neuron:

\bar{I}_{ik} = p(k)s\frac{N_{\text{post}} - 1}{N_{\text{pre}}},
I¯ik=p(k)sNpost−1Npre,\bar{I}_{ik} = p(k)s\frac{N_{\text{post}} - 1}{N_{\text{pre}}},

Average number of connections in network

Mean-field results

Cost per unit length for dendrite to cover an area

Set average number of connections to the cost for the dendritic tree

\frac{p(k)s}{\epsilon} N_{\text{post}} = N_{\text{pre}}^{2/3},
p(k)sϵNpost=Npre2/3, \frac{p(k)s}{\epsilon} N_{\text{post}} = N_{\text{pre}}^{2/3},
C = \epsilon\sqrt{N_{\text{pre}}},
C=ϵNpre,C = \epsilon\sqrt{N_{\text{pre}}},
p(k)s\frac{N_{\text{post}}}{N_{\text{pre}}} = \epsilon\sqrt{N_{\text{pre}}},
p(k)sNpostNpre=ϵNpre, p(k)s\frac{N_{\text{post}}}{N_{\text{pre}}} = \epsilon\sqrt{N_{\text{pre}}},
\bar{I}_{ik} = C,
I¯ik=C,\bar{I}_{ik} = C,
N_{\text{pre}} \propto N_{\text{post}}^{3/2},
Npre∝Npost3/2,N_{\text{pre}} \propto N_{\text{post}}^{3/2},

Bayesian inference

BF_{01} = 4.7
BF01=4.7BF_{01} = 4.7

Bayesian inference

Dendritic pruning as an age-dependent branching process

Cowan, (1984), Azevedo & Leroi (2000) PNAS

\mu = \theta \cdot e^{\alpha\cdot t}
μ=θ⋅eα⋅t\mu = \theta \cdot e^{\alpha\cdot t}
\sigma^2 = \lambda \theta^2 \cdot e^{2\alpha\cdot t} = \mu^2
σ2=λθ2⋅e2α⋅t=μ2\sigma^2 = \lambda \theta^2 \cdot e^{2\alpha\cdot t} = \mu^2

Data you shouldn't
go to war on

Data you do
go to war on

Thank you!

scRNA-seq

Gene specificity

about ~24'000 genes expressed in the brain. 

\text{Let us define the following variables.}
Let us define the following variables.\text{Let us define the following variables.}
c : \text{a unique single cell.} \quad \text{Where: } c \in \{1, ... , C\}, \text{ and } C = 380.
c:a unique single cell.Where: c∈{1,...,C}, and C=380.c : \text{a unique single cell.} \quad \text{Where: } c \in \{1, ... , C\}, \text{ and } C = 380.
m : \text{a unique single gene.} \quad \text{Where: } m \in \{1, ... , M\}, \text{ and } C = 380.
m:a unique single gene.Where: m∈{1,...,M}, and C=380.m : \text{a unique single gene.} \quad \text{Where: } m \in \{1, ... , M\}, \text{ and } C = 380.
D : \text{a } C \times D \text{ read count matrix.}
D:a C×D read count matrix.D : \text{a } C \times D \text{ read count matrix.}
D_{c,m} : \text{ normalized number of reads mapped to cell } c \text{ for gene } m.
Dc,m: normalized number of reads mapped to cell c for gene m.D_{c,m} : \text{ normalized number of reads mapped to cell } c \text{ for gene } m.
r_{c,m} : \text{ cell-gene specificity ratio.}
rc,m: cell-gene specificity ratio.r_{c,m} : \text{ cell-gene specificity ratio.}
r_{c,m} = \frac{D_{c,m}}{ s_m }
rc,m=Dc,msmr_{c,m} = \frac{D_{c,m}}{ s_m }
s_{m} = \frac{1}{C} \sum_{i=1}^C D_{i,m}
sm=1C∑i=1CDi,ms_{m} = \frac{1}{C} \sum_{i=1}^C D_{i,m}
s_{m} : \text{ gene specificity.}
sm: gene specificity.s_{m} : \text{ gene specificity.}
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