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FORSCHUNG

Nonrandom network connectivity comes in pairs

Felix Z. Hoffmann1,2 and Jochen Triesch1

1Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe University, Frankfurt am Main, Deutschland
2International Max Planck Research School for Neural Circuits, Max Planck Institute for Brain Research,
Frankfurt am Main, Deutschland

Schlüsselwörter: Nonrandom connectivity, Cortical circuit, Bidirectional connections, Random
graph model

ABSTRAKT

Overrepresentation of bidirectional connections in local cortical networks has been
repeatedly reported and is a focus of the ongoing discussion of nonrandom connectivity.
Here we show in a brief mathematical analysis that in a network in which connection
probabilities are symmetric in pairs, Pij
and nonrandom structures are inherently linked; an overabundance of reciprocally
connected pairs emerges necessarily when some pairs of neurons are more likely to be
connected than others. Our numerical results imply that such overrepresentation can
also be sustained when connection probabilities are only approximately symmetric.

= Pji, the occurrences of bidirectional connections

ZUSAMMENFASSUNG DES AUTORS

Understanding the speciﬁc connectivity of neural circuits is an important challenge of
modern neuroscience. In this study we address an important feature of neural connectivity,
the abundance of bidirectionally connected neuron pairs, which far exceeds what would be
expected in a random network. Our theoretical analysis reveals a simple condition under
which such an overrepresentation of bidirectionally connected pairs necessarily occurs: Any
network in which both directions of connection are equally likely to exist in any given pair of
Neuronen, but in which some pairs are more likely to be connected than others, must exhibit
an abundance of reciprocal connections. This insight should guide the analysis and
interpretation of future connectomics datasets.

Increasing evidence is showing the highly structured nature of cortical microcircuitry
(Perin, Berger, & Markram, 2011; Song, Sjöström, Reigl, Nelson, & Chklovskii, 2005). Not ev-
ery connection is equally likely to be established, but some pairs of neurons are more likely
to connect than others. In diesem Kontext, the relative occurrence of bidirectionally connected
pairs has been of particular interest. Using data obtained from paired whole-cell recordings in
cortical slices, the amount of bidirectionally connected pairs was compared with the number
of reciprocal pairs that one would expect in a random network with the same overall con-
nection probability. The connectivity of Layer 5 pyramidal neurons in the rat visual cortex
(Song et al., 2005) and somatosensory cortex (Markram, Lübke, Frotscher, Roth, & Sakmann,
1997; Perin et al., 2011) was shown to have signiﬁcantly more reciprocal connections than
expected.

The prevalence of bidirectional connectivity has since been established as an important
indicator of the nonrandomness of a network (Lefort, Tomm, Floyd Sarria, & Petersen, 2009;

Keine offenen Zugänge

Tagebuch

Zitat: Hoffmann, F. Z., & Triesch J.
(2017). Nonrandom network
connectivity comes in pairs.
Netzwerkneurowissenschaften, 1(1), 31–41.
doi:10.1162/netn_a_00004

DOI:
http://doi.org/10.1162/netn_a_00004

zusätzliche Informationen:
doi.org/10.6084/m9.ﬁgshare.3501860
doi.org/10.5281/zenodo.200368
https://non-random-connectivity-
comes-in-pairs.github.io/

Erhalten: 16 September 2016
Akzeptiert: 19 Dezember 2016

Konkurrierende Interessen: Die Autoren haben
erklärte, dass keine konkurrierenden Interessen bestehen
existieren.

Korrespondierender Autor:
Felix Z. Hoffmann
hoffmann@ﬁas.uni-frankfurt.de

Handling-Editor:
Olaf Sporns

Urheberrechte ©: © 2017
Massachusetts Institute of Technology
Veröffentlicht unter Creative Commons
Namensnennung 4.0 International
(CC BY 4.0) Lizenz

Die MIT-Presse

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Nonrandom network connectivity comes in pairs

Cortical microcircuitry:
The set of neuron-to-neuron
connections in a small subset of cells
in the cortex.

Support:
The subset of the domain of a
real-valued function whose elements
are not mapped to zero.

Erd ˝os–Rényi graph:
Randomly wired network in which
each possible connection has the
same probability to exist.
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Bourjaily & Müller, 2011). Jedoch, the exact relationship between nonrandomness and rela-
tive reciprocity has not been explained. Hier, we model cortical circuitry as random networks
in which each possible connection has a separate probability to exist. Using this model, we are
able to show that any nonrandom connectivity, expressed as higher connection probabilities
in some edges and lower probabilities in others, necessarily induces a relative overrepresen-
tation of bidirectional connections, as long as the connection probabilities remain symmetric
within pairs. Quantitatively, we analyze reciprocity in networks with one discrete and one
continuous distribution of connection probabilities to demonstrate that the relative abundance
of bidirectional connections reported in experimental studies can easily be obtained from
these models.

ERGEBNISSE

The emergence of nonrandom connectivity patterns can be modeled by assigning each pos-
sible connection in a random graph a separate probability to exist.
In such a model some
connections are more likely to be realized than others, allowing for the encoding of patterns
within the speciﬁc probabilities of each connection.
In the limiting case, each connection
either exists or is absent with certainty, representing a blueprint for the network architecture.

To analyze the effect of nonrandom structures within a network—speciﬁcally, on the statis-
tics of bidirectionally connected pairs found in the network—we consider a random graph
model of N neurons in which the probability of node i to connect to node j is modeled
by a random variable Pij. Für diese Analyse, we assume Pij, for i, j = 1, . . . , N with i (cid:2)= j,
to be identically distributed random variables in [0, 1], yielding a probability of connection
for each ordered pair of nodes in the graph. Outside of pairs, the random variables Pij are
assumed to be independent—that is, nonequal Pij and Pkl are independent as long as i (cid:2)= l
(cid:2)= k. Endlich, we explicitly exclude self-connections in this model and assume at all
or j
times that i (cid:2)= j.

Given the distributions of connection probabilities, what is then the probability in this model
for a randomly selected node to have a projection to another randomly selected node? Seit
the random variables Pij are identically distributed, we compute this overall connection prob-
ability μ easily as the expected value of Pij,

μ = E(Pij

(1)

Zum Beispiel, if the values of Pij have a probability density function f with support in [0, 1], Wir
can compute the connection fraction as

μ =

(cid:2)

x f (X) dx.

(2)

In this work, we are interested in the probability Pbidir that a bidirectional connection exists
in a random pair of neurons. We determine Pbidir as the expected value of the product of Pij
and Pji,

Pbidir

= E(PijPji

(3)

The relative occurrence (cid:2) of such reciprocally connected pairs compares Pbidir with the
occurrence of bidirectional pairs in an Erd ˝os–Rényi graph, in which each unidirectional
connection is equally likely to occur with probability μ (Erd ˝os & Rényi, 1959; Gilbert, 1959).

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Nonrandom network connectivity comes in pairs

The probability that a particular bidirectional connection exists in such a random graph is
simply μ2, and we obtain the relative occurrence as the quotient

(cid:2) = Pbidir
μ2

)
= E(PijPji
(cid:4)
(cid:3)
2 .
Pij
E

(4)

Experimental studies in local cortical circuits of rodents have repeatedly reported a rela-
tive occurrence of bidirectional connections (cid:2) > 1 (Markram et al., 1997; Perin et al., 2011;
Song et al., 2005). To understand in which cases such an overrepresentation occurs, we con-
sider two cases. In the ﬁrst case, assume that connection probabilities are independently de-
termined within pairs, sowie, meaning that the random variables Pij and Pji are independent.
Dann, because Pij and Pji are identically distributed,

E(PijPji

) = E(Pij

) E(Pji

) = E(Pij

)2,

(5)

and we would expect to observe no overrepresentation of reciprocal connections, (cid:2) = 1. In
= Pji.
the second case, assume that connection probabilities are symmetric with in pairs, Pij
In this case,

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and the expected relative occurrence of reciprocal connections becomes

Pbidir

= E(P2
ij

(cid:2) =

)
E(P2
ij
(cid:3)
(cid:4)
2 .

Pij

(6)

(7)

Jensen’s inequality:
If ϕ is a convex function and X an
integrable random variable, Dann
ϕ(E(X)) ≤ E(ϕ(X)), where E(X)
denotes the expected value of X.

We note that now any distribution of Pij with a nonvanishing variance will lead to a relative
occurrence that deviates from the Erd ˝os–Rényi graph, Weil

Var(Pij

) = E(P2
ij

) − E

(cid:3)

(cid:4)

Pij

(8)

Darüber hinaus, since x (cid:3)→ x2 is a strictly convex function, Jensen’s inequality (Cover & Thomas,
2006; Jensen, 1906) yields

E(P2
ij

) ≥ E(Pij

)2,

(9)

and we ﬁnd that (cid:2) ≥ 1 in networks with symmetric connection probabilities. Jensen’s inequal-
ity further states that the equality in (9), and thus (cid:2) = 1, holds if and only if Pij follows a
degenerate distribution—that is, if all Pij take the identical value μ. In the other case, Wo
Pij takes on more than one value with nonzero probability, we speak of a nondegenerate
distribution.

As a central result of this study, we thus ﬁnd that any nondegenerate distribution of sym-
= Pji) necessarily induces an overrepresentation of bi-
metric connection probabilities (Pij
directional connections in the network, (cid:2) > 1. Mit anderen Worten, in a network in which both
directions of connection are equally likely within any given pair, but in which some pairs
are more likely to be connected than others, the count of expected reciprocally connected
pairs is strictly underestimated by the statistics of an Erd ˝os–Rényi graph with the same overall
connection probability E(Pij

) = μ.

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Nonrandom network connectivity comes in pairs

Upper Bound for (cid:2)
The overrepresentation of bidirectional connections (cid:2) in a network is maximal when every
connected pair is already a reciprocally connected pair. In terms of the model deﬁned above,
this is the case when

The relative occurrence of reciprocal connections from (4) then becomes

E(PijPji

) = E(Pij

(cid:2) = 1
E(Pij

)

= 1
M .

(10)

(11)

Daher, for local cortical circuits of Layer 5 pyramidal neurons with a typical connection prob-
ability of μ = 0.1 (Song et al., 2005; Thomson, Westen, Wang, & Bannister, 2002), the network
model yields a maximal overrepresentation of (cid:2) = 10. While this theoretical maximum is
unlikely to exist in actual cortical networks, the precise degree of overrepresentation will
depend on the speciﬁc distribution of connection probabilities in the network. Im Folgenden
Abschnitte, we study two generic examples.

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Orientation preference:
The property of neurons in primary
visual cortex to respond strongly to
visual stimuli with a certain
orientation appearing inside their
receptive ﬁelds.

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Two-Point Distribution

The simplest nondegenerate distribution of connection probabilities is a distribution that takes
two values x, j, with probabilities p and 1 − p, jeweils, as illustrated in Figure 1A. Das
distribution may be seen as a crude approximation to the connection probabilities recently
observed in visual cortex as a function of the neurons’ absolute difference in orientation prefer-
enz, where a “high” connection probability was reported for a difference between 0° and 45°
and a “low” probability was seen for cells with a difference of 45
in orientation tuning
(Lee et al., 2016).

–90

◦

Formally, let x, y ∈ [0, 1] with x > y and 0 < p < 1. A random variable X follows the two-point distribution T (p, x, y) if P(X = x) = p and P(X = y) = 1 − p. In our network model, let then the Pij be T (p, x, y) distributed. The overall connection probability μ is μ = E(Pij ) = px + (1 − p)y. (12) Assume again that Pij = Pji. The relative occurrence of bidirectional connections is given by Solving (12) for p as (cid:2) = ) E(P2 ij μ2 = px2 + (1 − p)y2 μ2 . p = μ − y x − y (13) (14) and inserting this result into (13) yields an expression for the relative overrepresentation that depends on x, y, and μ (see Hoffmann & Triesch, 2016c, SI1), (cid:2) = x + y μ − xy μ2 . (15) 34 Nonrandom network connectivity comes in pairs Figure 1. Relative overrepresentation (cid:2) of bidirectional connections in networks with a fraction of pairs connected with a high probability x and the rest of the pairs connected with a low probability y. (A) Diagram illustrating the targets with a high chance x of connecting (thick arrows) and targets with a low probability y of connecting (thin arrows) for a single source node (hatched). (B) Different pairings of x and y can induce a high relative overrepresentation (cid:2) in a network with two-point- x−y , x, y), and a ﬁxed overall connection probability distributed connection probabilities, Pij μ = 0.1. The dashed line marks an overrepresentation of bidirectional connections of (cid:2) = 4, observed for Layer 5 pyramidal neurons in rat visual cortex (Song et al., 2005). ∼ T ( μ−y Here we ﬁx μ = 0.1 in accordance with the overall connection probability found in local circuits of pyramidal cells in the rat visual cortex (Song et al., 2005) and obtain the relative occurrence dependent on the two connection probability values x and y. Given x ≥ μ, it follows that y ≤ μ (see Hoffmann & Triesch, 2016c, SI2) and that the possible values for x and y are 0.1 ≤ x ≤ 1 and 0 ≤ y ≤ 0.1. Figure 1B shows contours of (cid:2) for these (x, y) pairings, illustrating how different values for the relative overrepresentation of reciprocal connections can be induced by two-point-distributed connection probabilities. We ﬁnd that in such net- works, higher values of (cid:2) are easily obtained with reasonable network conﬁgurations. For example, a relative overrepresentation of (cid:2) = 4 could be achieved by a two-point distribution of connection probabilities in which one group of neuron pairs is highly connected with prob- ability x = 0.7, while the other group of neuron pairs is sparsely connected with probability y = 0.05. Collectively, the highly connected pairs then make up less than 8% of all neuron pairs, showing that it is sufﬁcient to have a small subgroup of highly connected neuron pairs to induce a high overrepresentation of bidirectionally connected pairs in the network. For more densely connected networks, μ > 0.1, the effect that two distinct connection probabilities have
on the overrepresentation of reciprocal connections is reduced (vgl. Figure S1), as one would
intuitively expect from the dependence of the maximal overrepresentation on μ in (11).

Gamma Distribution

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Nächste, we analyze the relative overrepresentation of bidirectional connections in a network
with continuously distributed connection probabilities. The gamma distribution Γ(α, β) mit
probability density function

⎧
⎨

fα,β(X) =

1
βαΓ(α) X

α−1 e

−x/β

x ≥ 0,

⎩

ansonsten,

(16)

allows for the variance Var(X) = αβ2 of a gamma-distributed random variable X ∼ Γ(α, β),
to take on different values, while keeping its mean E(X) = αβ constant (Hogg & Craig, 1978).
The exponential distribution emerges as a special case of the gamma distribution (α = 1).

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Nonrandom network connectivity comes in pairs

To ensure that the randomly drawn connection probabilities lie within the interval [0, 1], Wir
here consider a modiﬁcation to the traditional gamma distribution, in the form of a truncated
Ausführung. Let α, β > 0. A random variable X follows the truncated gamma distribution ΓT(α, β)
if it has the probability density function

⎧
⎨

⎩

α,β(X) =
f T

Kα,β

1
βαΓ(α) X

α−1 e

−x/β

0 ≤ x ≤ 1,

ansonsten.

(17)

The factor Kα,β is the inverse of the cumulative probability that x ≤ 1 in the untruncated gamma
distribution,

and is needed to ensure that

Kα,β =

(cid:8)(cid:2)

(cid:9)−1

fα,β(X) dx

(cid:2)

α,β(X) dx = 1.
f T

(18)

(19)

Consider the above network model in which the connection probabilities PT
distributed and PT
ij

ji . We compute the relative overrepresentation (cid:2) numerically from

= PT

ij are ΓT(α, β)

(cid:10)

μ = E
(cid:10)

PT
ij

(cid:11)

(cid:2)

x f T

α,β(X) dx,

x2 f T

α,β(X) dx.

(20)

(21)

Pairings of the shape parameter α and the scale parameter β are chosen such that the over-
all connection probability reﬂects connectivity statistics in local cortical networks, μ = 0.1
(Song et al., 2005; Thomson et al., 2002). The probability density functions and the resulting
relative overrepresentation of reciprocal connections (cid:2) for four representative α, β pairs are
shown in Figure 2A. Hier, β is determined so as to yield μ = 0.1 for the given α, following the
relationship shown in Figure 2B (solid curve).

= 1
In the sparse networks we modeled, the tail of the gamma distribution is near zero at PT
ij
(see Figure 2A). Thus Kα,β ≈ 1, and the truncated gamma distribution can be well approx-
imated by the untruncated version. Assuming the connection probabilities to be standard
gamma distributed, Pij

∼ Γ(α, β), we have

E(P2
ij

) = Var(Pij

) + E(Pij

)2 = αβ2 + α2β2,

and thus

(cid:2) =

(cid:11)

(cid:10)

PT
ij

(cid:10)

(cid:3)

Pij

≈

Pij

(cid:11)

2
(cid:4)

α2β2
α2β2

αβ2
α2β2

= 1 + 1
α

=: ˜(cid:2).

(22)

(23)

The approximation (cid:2) ≈ ˜(cid:2) = 1 + 1

α works well for α ≥ 1, as is shown in Figure 2C.

To induce a high overrepresentation of reciprocal pairs in the network, the gamma distri-
bution of connection probabilities takes a highly skewed shape. To obtain (cid:2) = 4, nur 57% von
pairs are expected to have a higher connection probability than 0.01 (α = 0.248, β = 0.487).
Such a situation, in which a large part of all neuron pairs have a small connection probability

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Nonrandom network connectivity comes in pairs

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Figur 2. Relative occurrences of bidirectional connections (cid:2) in networks with gamma-distributed
(A) Probability density functions of the truncated gamma distribution
connection probabilities.
ΓT(α, β) for different shape parameters α and the induced relative overrepresentation (cid:2) in a net-
work with such distributed connection probabilities Pij. For a given α, the scale parameter β was
chosen such that μ = 0.1. The plot to the right continues the density functions at a different scale.
(B) Contours of α, β pairings that yield an overall connection probability of μ = 0.1. The dashed
line shows the approximation β = μ
α , where μ = 0.1. (C) Relative occurrences (cid:2) as a function of α
for ﬁxed μ = 0.1. For α ≥ 1, this relationship is well approximated by (cid:2) ≈ 1 + 1
α .

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while some few pairs have a high chance to be connected, is likely to occur if, zum Beispiel,
the connection probability strongly depends on the spatial separation of the neurons, as was
found in Layer 5 excitatory circuits of the rat somatosensory cortex (Perin et al., 2011). Dann
only nearby neurons are likely to be connected, while the larger part of more distant neurons
has a low probability of connection.

Symmetry of Connection Probabilities in Neural Circuits

In neural circuits, connection probabilities that are equal within pairs but differ across the net-
work are plausible from both an anatomical and a functional perspective. From the anatom-
ical point of view, the distance dependency of connection probabilities mentioned above
is a characteristic of cortical circuits that necessarily leads to symmetric probabilities: Der
distance from the ﬁrst neuron’s soma to the second neuron’s soma is the same as the distance
from the second to the ﬁrst, resulting in equal probabilities within a pair of neurons when
interneuron distance determines the connection probabilities. Regarding the functional per-
spective, connection probabilities may also depend on the functional properties of the cells
in the network. Zum Beispiel, the probability of connections between orientation-tuned cells
in the mouse primary visual cortex depends on their absolute difference in orientation tuning
(Ko et al., 2011; Lee et al., 2016). Since the absolute difference in orientation tuning will be
the same in both directions, connection probabilities can be expected to be equal within a
pair of orientation-tuned cells.

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Nonrandom network connectivity comes in pairs

Jedoch, even when connection probabilities within pairs do not match exactly, an over-
representation of reciprocal connections is still likely to be observed when connection proba-
bilities follow a nondegenerate distribution. To see this, consider that connection probabilities
(X). As before, we as-
Pij are distributed according to some probability density function fPij
sume that the values of Pij are independent outside of pairs. In the following discussion, Wir
also assume that i > j without loss of generality. The expected probability of a reciprocal
connection within a pair can then be expressed as

E(PijPji

) =

(cid:2)

xy fPij,Pji

(X, j) dx dy,

where fPij,Pji

(X, j) is the joint probability density function of Pij and Pji,

fPij,Pji

(X, j) = fPji

|Pij

(j | X) fPij

(X).

(24)

(25)

= Pji, fPji

In the case that Pij and Pji are independent, we have fPji
of Pij
function that transitions between the two extreme cases by multiplying fPji
function of a normal distribution centered around x,

(j), and in the case
(j | X) = δ(y − x). Here we propose a model for the conditional density
(j) by the density

(j | X) = fPji

|Pij

fPji

|Pij

(j | X) = 1

Nσ(X) fPji

(j)

1
√
2π

(y−x)2
2σ2

where the additional factor Nσ(X)−1 makes sure that fPji

|Pij

(j | X) integrates to 1,

Nσ(X) =

(cid:2)

(z)

fPji

1
√
2π

(z−x)2
2σ2 dz.

(26)

(27)

In der Tat, as the standard deviation σ of
fPji

(j | X) approaches fPji

|Pij

(j), and in the limit σ → 0 we have

the modulating normal distribution increases,

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lim
σ→0

fPji

|Pij

(j | X) = δ(y − x).

(28)

In Figure 3A, the conditional density functions for various σ are shown for the truncated
gamma distribution. For low values of σ, the conditional density function resembles a narrow
Gaussian around x, reﬂecting approximately symmetric connection probabilities. For σ > 1,
α,β(j), reﬂecting
andererseits, fPji
the independence of Pji from Pij.

(j | X) becomes virtually indistinguishable from f T

|Pij

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Endlich, we employ the model to examine how the relative overrepresentation of bidirec-
tional connections (cid:2) changes with the degree of symmetry in the connection probabilities
within a pair of neurons. To do so, (cid:2) is computed as a function of σ for a given distribution of
Pij using

(cid:2) = E(PijPji
μ2

)

(29)

the numerator is deﬁned through (24–26), and the overall connection probability μ is calcu-
lated as

μ = 1
2

(cid:2)

x fPij

(X) dx + 1
2

(cid:2)

(X)

fPij

(cid:2)

y fPji

|Pij

(j | X) dy dx.

(30)

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Nonrandom network connectivity comes in pairs

|Pij

(j) = f T

α,β(j), with α = 0.248 and β such that E(Pij

Figur 3. Relative overrepresentation of bidirectional connections (cid:2) is sustained when connection
probabilities are only approximately symmetric in pairs. (A) Illustration that the conditional density
(j | X) In (26) transitions from equality of the random variables Pij and Pji to indepen-
function fPji
) = 0.1.
dence with increasing σ. We use fPij
For the illustration, Pij was ﬁxed as x = 0.15. Already for σ = 1, the conditional density function
α,β(j) (vgl. Figure 2A). (B) Relative occurrences of recip-
becomes visually indistinguishable from f T
rocally connected pairs (cid:2) as a function of σ. The curves for α = 1 and α = 2 show numerical
) = 0.1. Relative recip-
solutions of (29) with fPij
rocal pair counts from generated networks following the model matched these theoretical curves
(data not shown). For α = 0.248, random variables with the respective probability density functions
were sampled and the average (cid:2) was computed via (29) using the sample means. Error bars show
SEMs; the curve for α = 0.248 (solid line) was ﬁtted to the data points and is purely for illustrative
Zwecke.

α,β(j), where β was chosen such that E(Pij

(j) = f T

|Pij

since half of the connection probabilities are drawn according to fPij and the other half are
drawn according to fPji
given a particular value of x drawn according to fPij. Figure 3B shows
the change of (cid:2) with σ for connection probabilities Pij following a truncated gamma distribu-
tion ΓT(α, β). For the three parameter sets chosen, we see that a strong overrepresentation of
bidirectional connections is sustained when connection probabilities are only approximately
symmetric in pairs. Außerdem, as long as Pji is at biased to take similar values to Pij, an over-
representation of (cid:2) > 1 can be observed, implying that effects such as distance dependency
or the dependence on the absolute difference in orientation tuning of connection probabilities
will tend to increase the relative occurrence of bidirectional connections, even when other
effects are also inﬂuencing the neurons’ connection probabilities.

DISKUSSION

Experimental evidence suggests that any pair of excitatory cells within a cortical column has
contact points between axon and dendrite close enough to support a synaptic connection
between the cells (Kalisman, Silberberg, & Markram, 2005; Stepanyants, Tamás, & Chklovskii,
2004). Despite this potential “all-to-all” connectivity, only a small fraction of the contacts are
realized as functional synapses. Uncovering the principles underlying which contact points get
utilized for synaptic transmission is crucial for our understanding of the structure and function
of the local cortical circuits in the mammalian brain.

Local networks in the rat visual and somatosensory cortices have been shown to feature non-
random structure (Perin et al., 2011; Song et al., 2005), and much attention has been given to
bidirectionally connected neuron pairs that occur more often than would be expected from
zufällige Konnektivität (Bourjaily & Müller, 2011; Clopath, Büsing, Vasilaki, & Gerstner, 2010;
Miner & Triesch, 2016).
In this study we have shown a condition under which nonrandom
network structure and the occurrence of reciprocally connected pairs are inherently linked;
a relative overrepresentation of bidirectional connections arises necessarily in networks with

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Nonrandom network connectivity comes in pairs

a nondegenerate distribution of symmetric connection probabilities. The absence of an over-
abundance of reciprocal pairs, andererseits, as for example is found in the intralayer
connectivity of the mouse C2 barrel column (Lefort et al., 2009), points toward either a truly
random network or an asymmetry in the connection probabilities.

Quantitatively, a network in which connection probabilities take on one of two values
is easily able to account for even the highest values of overrepresentation reported. A network
with such a two-point distribution of connection probabilities might occur naturally, Wo
the probability of connection depends on whether a given pair of neurons shares a certain
feature—for example, have or do not have similar orientation preferences (Lee et al., 2016).

A continuous distribution in connection probabilities, andererseits, might occur when
pair connectivity depends on a continuous parameter, such as the interneuron distance or the
neurons’ ages. We showed that networks in which connection probabilities follow a gamma
distribution can also have a high relative occurrence of reciprocally connected pairs; Jedoch,
in this case a larger fraction of pairs remain unconnected with a very high probability.

It is likely that a combination of such effects determines the connection probabilities in
local cortical networks. Wichtig, we showed that as long as this probability is symmetric
for pairs, any such effect that creates a nondegenerate distribution of probabilities will cause
an increase of the reciprocity in the network.

Our results conﬁrm the intuitive notion that reciprocity is favored in symmetric networks,
whereas asymmetric probabilities of connection inhibit the occurrence of bidirectionally con-
nected pairs. Network models with symmetric connectivity, such as Hopﬁeld nets, generally
excel at memory storage and retrieval through ﬁxed-point attractor dynamics (Hopﬁeld, 1982),
while asymmetric network models such as synﬁre chains are suitable for reliable signal trans-
mission (Abeles, 1982; Diesmann, Gewaltig, & Aertsen, 1999). This suggests the intriguing
possibility that one may be able to infer the nature of the computations in a neural circuit
on the basis of certain statistics of its connectivity, such as the abundance of bidirectionally
connected pairs.

Abschließend, the present study puts the overrepresentation of bidirectional connections
found in local cortical circuits in a new light.
If connection probabilities are symmetric in
pairs, the overrepresentation emerges as a symptom of any form of nonrandom connectiv-
ität.
It is thus crucial for both future experimental and modeling studies to develop a more
reﬁned view of nonrandom network connectivity that goes beyond simple pair statistics.
Focusing on higher-order connectivity patterns and taking into account the actual synaptic
efﬁcacies seem promising avenues for future research into the nonrandom wiring of brain
circuits.

SUPPLEMENTARY INFORMATION

The supplementary information document for references SI1 and SI2 and for Figure S1 is
available online at DOI:10.6084/m9.ﬁgshare.3501860. Python code for the numerical comp-
utations is available as a GitHub repository, archived along with the generated data at
DOI:10.5281/zenodo.200368. A website documenting the code can be found at https://
non-random-connectivity-comes-in-pairs.github.io/

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers for their helpful and constructive comments on
earlier versions of this article.

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Nonrandom network connectivity comes in pairs

BEITRÄGE DES AUTORS

J.T. is supported by the Quandt Foundation. F.Z.H. and J.T. conceived and designed the study.
F.Z.H. carried out the formal analysis and performed the simulations. And both F.Z.H. and J.T.
wrote the manuscript.

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