Original Article<EMAIL_ADDRESS>This material is based upon work
supported by the National Science Foundation under Grant 1651396.

# Circumstantial evidence and explanatory models for synapses in large-scale
spike recordings

Ian H. Stevenson

###### Abstract

Whether, when, and how causal interactions between neurons can be meaningfully
studied from observations of neural activity alone are vital questions in
neural data analysis. Here we aim to better outline the concept of functional
connectivity for the specific situation where systems neuroscientists aim to
study synapses using spike train recordings. In some cases, cross-correlations
between the spikes of two neurons are such that, although we may not be able
to say that a relationship is causal without experimental manipulations,
models based on synaptic connections provide precise explanations of the data.
Additionally, there is often strong circumstantial evidence that pairs of
neurons are monosynaptically connected. Here we illustrate how circumstantial
evidence for or against synapses can be systematically assessed and show how
models of synaptic effects can provide testable predictions for pair-wise
spike statistics. We use case studies from large-scale multi-electrode spike
recordings to illustrate key points and to demonstrate how modeling synaptic
effects using large-scale spike recordings opens a wide range of data analytic
questions.

###### keywords:

spike trains, cross-correlation, synapses, connectivity

##  1 Introduction

In many cases, just by chance, large-scale spike recordings will happen to
sample pairs of neurons that are monosynaptically connected. The mouse (mus
musculus) brain, for instance, has $\sim$ 71 million neurons with $\sim$8
thousand synapses per neuron [14, 41]. Assuming that neurons are uniformly
randomly connected, this would imply that 1 out of every $\sim$9 thousand
pairs of neurons are monosynaptically connected. Although this is a small
proportion, in a large-scale recording of 1000 neurons ($\sim$1 million
possible connections) one could expect to observe $\sim 100\pm 10$
monosynaptically connected pairs just by chance (mean $\pm$ s.d.). Since
electrode arrays often record from a single brain region or a few adjacent
brain regions targeted for their connectivity, connection probabilities are
typically much higher in practice. Estimates of connection probabilities from
intracellular recordings in cortical slices suggest that for an electrode
spacing <100$\mu m$ the chances of two neurons being connected are on the
order of 1 in 10 [45, 87, 96]. Although the number of strongly connected pairs
in a typical large-scale spike recording may be small relative to the number
of possible pairs, sampling at least some connected pairs will become
inevitable as the number of neurons that can be simultaneously recorded
increases [17, 99]. Accurately detecting these connections from sparse spiking
activity alone is a statistical challenge, but identifying connected pairs can
allow us to characterize neural circuits in action.

Evidence for synaptic connections can, in many cases, appear in the cross-
correlation between the spike trains of two neurons [29, 34, 69, 75]. For a
strong monosynaptic connection, the spiking of the presynaptic neuron affects
the probability of postsynaptic spiking in the short time period following
presynaptic spikes. All things being equal, an excitatory synaptic connection
will increase the postsynaptic spiking probability, while an inhibitory
synaptic connection will decrease the postsynaptic spiking probability,
relative to the postsynaptic neuron’s baseline firing rate. [59] introduced
two key metrics for assessing spike transmission: efficacy and contribution.
The synaptic efficacy measures the excess probability of the postsynaptic
neuron spiking following a presynaptic spike, while contribution measures the
proportion of postsynaptic spikes considered to be the result of presynaptic
spikes. For excitatory synapses, efficacy and contribution are both between 0
and 1, and there are measurements of these quantities for many neural systems.
Usrey and colleagues, for instance, found efficacies ranging 0.6$\%$ to 36$\%$
for retinogeniculate synapses [115] and 0.1$\%$ to $\sim 30\%$ for
thalamocortical synapses [112] in cats. Swadlow and colleagues have found
similar efficacies in rabbit thalamocortical projections [123] with some
powerful, divergent connections having efficacies up to 20$\%$ [106]. Putative
synapses in the early auditory pathway have strong connections with both
efficacies and contributions between 1-10$\%$ [67, 121], and calyceal synapses
in the mammalian auditory midbrain can even have efficacies >50$\%$ [37, 48].
With large-scale recordings, [28] found efficacies from 0.1$\%$ to $\sim 20\%$
in the hippocampus, while [92] found median efficacies of $\sim 3\%$ for
retinal projections to superior colliculus. These examples provide only a
snapshot of a few neural systems, but they illustrate a fundamental point:
single presynaptic inputs can modify postsynaptic spiking probability by
several percentage points in typical cases and up to 10’s of percentage points
in extreme cases. Synaptic efficacies are, thus, often large enough that
individual synapses can be studied using spike observations.

Although our goal is to estimate causal synaptic effects [42], there are
limits to what we can conclude about synapses from spike observations alone.
It is important to note that we generally cannot infer the presence of a
connection with complete confidence from only the observed cross-correlation
[15, 33, 101] or rule out the possibility of a connection based on the absence
of a correlation. Interpreting observed cross-correlation (or other measure of
“functional connectivity”) as a direct estimate of causal impact would be a
mistake [66]. There are experimental methods for empirically verifying the
presence of synapses, such as antidromic stimulation, juxtacellular/micro-
stimulation, optogenetic stimulation, or by applying synaptic blockers [28,
90, 92, 107]. Without such manipulations or without being able to otherwise
control for possible confounds [58], efficacy estimates are unverified,
indirect, and potentially biased estimates of causal synaptic effects.
However, even without verification, it is worth considering that, for some
recorded neurons, their spike timing is almost certainly impacted by the
presence of synapses with other recorded neurons. Models of synaptic effects
may be the most parsimonious explanation for rapid, transient changes in
postsynaptic spike probability following presynaptic spikes. Indirect or
“circumstantial” evidence from other sources, such as spike waveforms and
anatomical location, can also strengthen (or weaken) our belief that a given
pair of neurons is synaptically connected. In general, strong synaptic
connections will impact the timing, shape, stability, and dynamics of pairwise
spike statistics when they are present. Accurate models of neural circuits
should be able to explain these effects, and fitting models based on synaptic
connections may be fruitful, as long as we keep in mind that there are errors
and biases in synapse detection.

Acknowledging the limitations of modeling putative synapses from spikes, there
are many interesting scientific questions that we can at least begin to answer
using observation alone. How does synaptic strength change with the precise
timing of presynaptic activity? Are the strengths of synapses stable in vivo?
How do synaptic strengths vary with behavior or stimuli or during learning?
How might presynaptic spikes contribute to a postsynaptic neuron’s tuning
properties? How does spike transmission through specific microcircuits change
under different brain states? And many others. Although we may not be able to
completely address these questions without confirmation from intracellular or
other measurements, the efficacies of putative synapses provide initial
(potentially biased) estimates of causal synaptic effects, and models based on
synaptic effects provide potential explanations that can be tested and
compared against alternative models. Previous studies have aimed to describe
the detailed shape of spike cross-correlations at putative synapses [30, 43,
44, 78], the dynamics of membrane potential integration [19], short-term
plasticity [28, 35, 36], behavioral and brain state dependencies [22, 31, 88,
102], and receptive field/tuning curve construction [3, 64, 65, 67, 83]. In
each of these studies, models of synaptic effects generate quantitative
predictions and act as key tools to interpret the observed correlations.

Here we provide an overview of methods and logic for modeling synapses from
large-scale spike train observations. Although the specific models and
concepts here have been previously described, the aim of this paper is to
illustrate how these varied approaches can be synthesized into a coherent data
analysis framework. The results that follow focus on new case-studies from
experimental data that directly demonstrate key ideas. Interspersed with these
results, however, we discuss conceptual challenges and highlight previous
studies that introduced or expanded on these challenges studying synapses from
spikes. In the first section, we consider the basic theoretical problem of
when synapses can be detected from spike observations alone. This detection
problem is often framed as a hypothesis test, and a power analysis provides
useful “rules of thumb” for distinguishing when synapses can be detected and
when they cannot. Second, we discuss a conceptual framework for interpreting
synaptic effects beyond the hypothesis test using circumstantial evidence.
Third, after discussing circumstantial evidence broadly, we use case studies
to directly illustrate the impact of specific circumstantial evidence, such as
anatomy and spike waveforms. Fourth, we illustrate some of the ways that
existing model-based approaches can account for detailed patterns of pairwise
spike statistics and describe synaptic plasticity. Fifth, we discuss concerns
about confounding due to common input and polysynaptic effects. Lastly, we
illustrate how putative synapses detected in large-scale spike recordings are
reaching a scale where they might allow observing and characterizing putative
microcircuits.

Figure 1: Example putative synaptic connections detected from a multi-
electrode array spike recording of a slice culture of mouse somatosensory
cortex (CRCNS ssc3-23). A) Estimated network structure with single units
aligned to their spatial locations on the multi-electrode array. B) Example
cross-correlations from strong putative excitatory (purple) and inhibitory
(green) monosynaptic connections. Triangles denote the peak or trough of the
putative synaptic effect. C) Multiple putative synapses from the same
presynaptic neuron are often either all excitatory or all inhibitory,
consistent with Dale’s Law. Colors in (B) and (C) correspond to the edges
shown in (A).

##  2 Results

We start by considering the primary statistical evidence for the presence of a
synaptic connection: the cross-correlogram. Here the spiking of a putative
postsynaptic neuron is compared to the spikes of a putative presynaptic
neuron. Given the binned spiking of the two neurons $n_{pre}$ and $n_{post}$
over time, the cross-correlogram is

$c(\tau)=\sum_{t}{n_{pre}(t)n_{post}(t+\tau)}$

Increases or decreases in $c(\tau)$ at a delay $\tau$ between 0ms and
(approximately) 5-10ms may reflect synaptic effects when a synapse is present.
Detecting a short-latency, transient deviation in $c(\tau)$ relative to an
expected baseline provides the basis for many synapse detection methods [6,
69]. Typically, the observed count within a specific range of $\tau$ is
compared to a null hypothesis that describes what is expected to happen in the
absence of a synapse between the neurons. If the observed count is unlikely
under the null hypothesis, this provides some evidence that there may be
synaptic effect.

To give some initial, concrete examples of what putative synaptic effects look
like and a sense of how these effects are detected we introduce a case-study
(Fig 1). Here, we analyze spike data from an in vitro multi-electrode array
recording of mouse somatosensory cortex. There are $M=310$ neurons with
$M(M-1)$ possible (directed) connections, $\sim$100K in this recording. After
testing all possible pairs, we can construct a putative synaptic network for
the observed sample of units (Fig 1A). A small subset of pairs have a short-
latency transient increase in postsynaptic firing following presynaptic spikes
(Fig 1B, top), while others show a short-latency transient decrease (Fig 1B,
bottom). The latency, timescale, and efficacy of these patterns are often
consistent with expected synaptic effects.

###  2.1 Power analysis for the detection of synapses from spikes

Since an individual synapse represents only one of, potentially, many inputs
to a postsynaptic neuron, it is reasonable to ask: when can the impact of a
single input be reliably detected and measured from spikes alone? Detecting a
synaptic connection from a noisy correlogram can be framed as a null
hypothesis significance test, and, as with other hypothesis tests, power
analysis can be used to determine under what circumstances real effects are
expected to result in statistically significant outcomes. Given an effect
size, a sample size, and a confidence level for the hypothesis test $\alpha$,
we can calculate the probability of having a false negative (Type II error)
$\beta$. Or, conversely, given a sample size, confidence level, and desired
power, we can calculate the effect size necessary to achieve that power.

Let’s consider a simplified situation where we aim to detect a synaptic
connection with efficacy $e$ given $N$ presynaptic spikes and known
probability p for the postsynaptic neuron firing by chance within a fixed
detection window following each presynaptic spike. We wish to decide between
two hypotheses: $H_{0}:e=0$ versus $H_{A}:e\neq 0$. Assuming that the
observations are independent, the null distribution for the number of
postsynaptic spikes in the detection window is given by $y\sim Binomial(N,p)$
and the alternative distribution is $y\sim Binomial(N,p+e)$ (Fig 2A). Testing
in this situation is analogous to a one-sample test of a proportion, and the
power $(1-\beta)$ can be calculated exactly. Fig 2B illustrates how the power
varies with the effect size ($e$) and sample size ($N$). For experimentally
plausible values ($\sim$1000 presynaptic spikes), the power is often high
enough that even synapses with relatively low efficacy (5$\%$) could be
reliably detected.

Similarly, we can identify the minimal efficacy necessary to achieve a
specific power (Fig 2C). With some simplifying approximations, an approximate
threshold of detectable effects is $e\approx 1.4/\sqrt{N}$ (for the commonly
used values of confidence level $\alpha=0.05$ and power $0.8$, see Methods).
That is, to have an $\geq 80\%$ chance of detecting a synaptic connection
after observing 100, 1000, or 10000 presynaptic spikes, the efficacy would
need to at least 14$\%$, 4$\%$, or 1$\%$, respectively. This approximate power
analysis has important caveats; however, it provides a rule of thumb linking
the number of recorded spikes to the problem of synapse detection.

Several more complex hypothesis test-based methods for detecting putative
synapses [6, 31, 62, 97], as well as several model-based methods for detecting
synapses have been recently developed [27, 51, 84, 122]. These methods are
more involved than the simple hypothesis test presented above. The jitter
method [6], for instance, relies on an empirical null distribution generated
by adding timing noise to the spikes, and GLMCC [51] detects pairs based on a
model goodness-of-fit. Despite their differences, the power, false positive,
and false negative rates of these models can be estimated when the ground-
truth connectivity is known, such as in simulations with controlled synaptic
currents or potentials [50, 51, 69, 84, 117]. These methods give slightly
different putative synaptic networks and their results depend on the specified
confidence level, but there is generally agreement across methods for the
pairs where the strongest effects are present. These studies, along with
experimental results from identified synapses [28, 92], suggest that synapses
can indeed be detected in vivo from only spikes. However, weak synapses are
less likely to be detected than strong synapses, synapses between neurons with
low firing rates are less likely to be detected than those between neurons
with high firing rates, and excitatory synapses may not necessarily be
detected at the same rate as inhibitory synapses [2, 51, 74, 117].

Figure 2: Power analysis for detecting synaptic connections with known
baseline and unknown efficacy. A) Here we consider a simplified hypothesis
test between a null hypothesis and an alternative where a synapse increases
(bottom) or decreases (top) the probability of postsynaptic spiking by a
specific efficacy. We assume that the window of interest is already known, and
that we run our test on a single observed count of the postsynaptic spikes.
Vertical lines denote decision boundaries for a two-tailed hypothesis test. B)
Using a normal approximation to the Binomial distribution we can calculate the
power associated with a specific efficacy. Here curves show the power for
multiple baseline firing probabilities $p$ at $N=100$ (top) and $N=1000$
(bottom) presynaptic spikes. C) Using this approximation, we can also find the
efficacy necessary to achieve a desired power. Here we show efficacies need to
achieve a power of 80$\%$, with inhibitory (top) and excitatory (bottom)
synapses. Dots denote calculations from the normal approximation to the
binomial. Curves denote the first-order Taylor approximation. In all cases
shown here we assume a confidence level $\alpha$=0.05.

###  2.2 Evaluating circumstantial evidence for synaptic connections

We now have a statistical argument that many synapses can, in principle, be
detected from spikes when they are present, and a statistically significant
transient in the correlogram provides some direct evidence of a potential
causal effect. However, we should be cautious in concluding that a synapse is
present from observations of spikes alone. In the absence of
electrophysiological, optogenetic, or pharmacological manipulations to verify
that these are synaptic effects, we cannot be sure that short-latency,
transient changes in spiking probability are “caused” by a synapse. Here we
briefly review the indirect, but multiple, lines of evidence that can,
nonetheless, strengthen or weaken our belief that pairs of simultaneously
recorded neurons may be monosynaptically connected.

First, we consider the detailed shape of the cross-correlogram. Although many
studies of functional connectivity do not explicitly distinguish between
synaptic effects and other possible sources of fast correlated spiking [20,
46, 101], when studying putative synapses directly, researchers often aim to
measure the latency and time scale of the synaptic effect [105, 115]. Due to
the conduction velocity and synaptic delay, synaptic effects, when present,
will have a short, but non-zero latency. Then, due to the receptor dynamics
(AMPA, NMDA, GABAA, and/or GABAB), the timescale of the synaptic effect will
be limited to a short duration window following the presynaptic spikes. Both
experimental [30, 78] and theoretical [43, 44] studies have examined the
transformation between synaptic potentials/currents and postsynaptic spike
probability, and these results provide constraints on what is possible with
synaptic transmission. A transient increase in the correlogram with a latency
<10 ms and a duration of <3ms, such as those in Fig 1B, is consistent with
synaptic transmission. However, an increase with a latency of 20-50ms and a
duration of >20ms is somewhat inconsistent with our expectations for
monosynaptic transmission via ionotropic receptors for small mammals.

A second source of circumstantial evidence comes from anatomy and histology.
The probability of two neurons being monosynaptically connected depends,
crucially, on their locations. A neuron recorded in the retina is not likely
to be directly connected to a neuron recorded in the spinal cord, and
structural connectivity, in general, constrains conclusions about putative
synapses. In some cases, neurons are targeted for their receptive field
alignment [112, 123]. Using information about receptive field overlap, which
brain areas are connected to which [40, 55], or simply the distance between
the recorded neurons can additionally constrain the inference of synaptic
connections. Even within brain areas, intracellular measures often find that
connection probability decreases with distance and latency grows with distance
[87, 96]. Putative synapses detected from extracellular recordings often
mirror these trends [28], and the trends can be used to constrain our
detection [84].

Extracellular spike waveforms offer another important piece of circumstantial
evidence. Since cell types differ in their waveform characteristics [57, 109],
we may have clear expectations about the direction (increasing or decreasing
spike probability) of synaptic effects at individual putative connections even
before we examine the cross-correlogram. Putative presynaptic units with broad
spike waveforms, often pyramidal neurons, tend to have excitatory putative
synaptic effects, while units with narrow spike waveforms, often fast-spiking
interneurons, tend to have inhibitory effects [11]. More detailed approaches
to cell type classification using more complex waveform features or the full,
multi-channel spatio-temporal waveform pattern [24, 60, 89] are also being
actively developed.

Next, we consider the patterns of putative synapses beyond a single pair. In a
large-scale multi-electrode recording, we may have many putative synapses that
are detected, and the pattern across these many connections provides
circumstantial evidence for or against the individual pairs of neurons being
connected. Dale’s Law, that neurons are either excitatory or inhibitory [103],
provides an important example. A putative presynaptic neuron, if it is
excitatory, is expected to make excitatory connections with all its
postsynaptic targets, while an inhibitory presynaptic neuron should make
inhibitory connections. Despite growing evidence that co-transmission of
neurotransmitters can occur [110], we often find that Dale’s Law does hold
(approximately) when examining putative synapses (Fig 1C), and other details
about connection probabilities in local circuits [38, 96] could also be used.

Lastly, we consider examining spike statistics in more detail. The overall
cross-correlogram summarizes the pairwise spike statistics across the whole of
an experimental recording. In most cases, using all spikes that are available
will give us the greatest statistical power for detecting putative
connections. However, the overall cross-correlogram is only one limited
summary of the dependency between pre- and postsynaptic spiking. Analyzing the
data during specific time periods, behaviors, brain states, or specific
patterns of pre-synaptic activity can provide a richer understanding of the
pre-post spike dependencies. On the one hand, we expect that when a synapse is
present, certain features of the spike statistics should be stable no matter
how we look. The latency and duration of the synaptic effect, for instance,
are expected to be relatively stable across time and should not change
substantially across behaviors or brain states (although, see [9]). On the
other hand, we expect that other features of the spike statistics should
systematically vary if there is a synapse. The postsynaptic spike probability
may show evidence of nonlinearities due to membrane potential integration,
such as different probabilities of spiking following a burst vs an isolated
presynaptic spike, and, due to short-term synaptic plasticity, the efficacy of
a true synaptic connection is expected to increase or decrease rapidly,
depending on the pattern of presynaptic activity. These phenomena have both
been observed with putative synapses [19, 28, 105]. Importantly, however, both
the stability and variation of the correlogram reflect predictable
consequences of synaptic biophysics. When they are present, they may provide
additional support for our belief that a putative synapse is a genuine
synapse.

Figure 3: Parameters and stability of putative synaptic effects for CRCNS
SSC-3 dataset 23. A) Latency, time constant, and efficacy estimated for
10-minute segments of the 1-hour recording. Symbols denote the 6 putative
synapses shown in Fig 1B, with putative excitatory connections in purple and
putative inhibitory connections in green. Ellipses denote estimated mean and
covariance. B) Correlograms for the 10-minute segments of data, along with a
model fit (gray) that assumes fixed latency and time constant. C) Across all
putative synapses, there are consistent differences in the distances between
pairs, latencies, and time constants for excitatory and inhibitory effects.
Connection probability (inset) for both effects decreases with distance
between the pairs. Lines denote exponential fit within a Binomial regression
of observed counts to possible pairs at a given distance.

Descriptions of the pre-post spike statistics beyond the overall correlogram
also allow models of synaptic effects to be explicitly tested. After fitting a
synaptic model to one subset of the data, the model predictions can be
evaluated on other subsets. In previous work we found that detailed models of
short-term synaptic dynamics can predict responses to triplets of presynaptic
spikes [36], slow rate-dependent fluctuations in efficacy [85], and
differences across stimuli [36]. Multiple models can be compared directly, and
individual models of synaptic effects are falsifiable, in the sense that their
predictions may be wrong.

###  2.3 Case-studies evaluating putative synapses in large-scale spike
recordings

To illustrate how multiple lines of circumstantial evidence can fit together
we further examine results from two experimental datasets: 1) the 1hr
recording from organotypic slice cultures of mouse somatosensory cortex
introduced above (Collaborative Research in Computational Neuroscience SSC-3
dataset 23) and 2) a 2.7hr multi-region recording from awake mouse using
multiple Neuropixels arrays (Allen Institute – Visual Coding Neuropixels
electrophysiology session 715093703). SSC3-23 contains M=310 sorted, single
units (automatic clustering based on PCA followed by manual refinement) with
median firing rate 1.8 Hz [0.8, 3.7] (quartiles). ABI-715093703 contains 745
sorted, single units (SNR>2, Kilosort 2) with median firing rate 6.7 Hz [3.1,
10.7].

We exclude single-units with fewer than 1000 spikes (0.28 Hz, 6.7$\%$ in
SSC3-23, 0.10 Hz, 0.3$\%$ in ABI-715093703), since even moderate putative
connections with efficacies $\sim 4\%$ would be difficult to detect with so
few spikes. This leaves $M^{\prime}=289$ units for SSC3-23 and
$M^{\prime}=743$ units for ABI-715093703. Based on the approximate power
analysis described above, we then assess limits on detecting putative synapses
with 80$\%$ power. Assuming $\alpha=0.05$, and $\beta=0.2$ and a 10ms synaptic
window following presynaptic spikes to estimate p, the median efficacy
detection limit across all pairs is ±0.3$\%$ (ABI-715093703) and ±0.5$\%$
(SSC3-23). That is, in these relatively long recordings we expect efficacies
less than $\sim 1\%$ to be detectable for typical pairs of units.

Here, rather than comparing or optimizing detection, we want to highlight how
circumstantial evidence can shape interpretation of cross-correlation
observations. We first detect candidate pairs using a fast approximation to
the jitter method (jitter timescale of 2ms) and correct for multiple
comparisons using Benjamini-Hochberg step-up procedure with False Discovery
Rate of $\alpha=0.1$ (see Methods). In SSC3-23 this step identifies 347 pairs
out of 83K possible pairs (0.4$\%$, Fig 1A) as having statistically
significant fast correlations. This approach detects deviations from expected
slow correlations but does not identify effects that are consistent or
inconsistent with putative synapses. We thus fit the correlograms for
statistically significant pairs with a model that separates the correlogram
into a slow, baseline component and a fast, synaptic effect. Candidate
correlograms are often well-fit by such a model (median pseudo-$R^{2}$ is
0.74), and the estimated parameters are often consistent with synaptic
timescales (Fig 3). Since some candidate correlograms are not well fit and/or
not consistent with synaptic effects, we next restrict our analyses to more
specific parameter ranges. Here we focus on pairs with pseudo-$R^{2}>0.5$,
time constant<2ms, and latency between 0.1-6ms with 213 pairs meeting these
criteria (61$\%$ of candidates, 0.3$\%$ of all possible pairs).

Figure 4: Stability, anatomy, waveforms of putative synaptic effects of
ABI-715093703. A) Correlograms for three example putative synapses during
different behavior and visual stimuli. Blue and gray bars denote observed
count data, red curves denote a model fit that assumes fixed latency and time
constant (but variable baseline and efficacy) across stimuli. Left column is a
pair of units in CA1, middle column is a putative presynaptic unit in CA3 and
postsynaptic unit in DG, right column is a pair in VISp. B) Connectivity
matrix for the putative connections detected from this dataset (top), and
ratio of putative excitatory to inhibitory pairs for each projection (bottom).
C) Waveform and firing rate statistics for the putative presynaptic neurons.
Putative excitatory connections tend to be from presynaptic neurons with wider
waveforms and to have lower firing rates. Grayscale background denotes the
histogram of all units across all ABI recordings. Density plots along edges
denote kernel estimates for ABI-715093703 separated by putative excitatory
connections (red) and putative inhibitory connections (blue). Black triangles
denote the example pairs in (A). Green and purple triangles denote the example
pairs in (D). D) Correlograms for example pairs with inconsistent waveform and
correlogram information (pairs are in CA1 top, VISpm middle, and VISp bottom).

Having latencies and time constants consistent with synaptic effects may be
sufficient to label these pairs of neurons “putative synaptic connections”,
but additional evidence can help us answer “how putative are we talking?” and
“is this pair worth studying more?”. The stability of the putative synaptic
effect is one piece of circumstantial evidence that can inform both questions.
Here, to assess the stability of the effects, we split the data into 10-minute
windows and estimate synaptic parameters for each segment of the data. We find
that, in many examples, latency and duration both tend to be reliable over
time (Fig 3A) with estimates varying by <1ms across segments. Efficacy, as
well as the baseline postsynaptic firing rate, can vary more substantially (on
the order of ±50$\%$), but typically the sign of the putative synaptic effect
does not change (Fig 3B). Variation in the correlograms over time can be well
described by a model that assumes that the latency and time constant of the
fast effect are fixed, while the efficacy and baseline vary.

In deciding whether a set of putative pairs is worth studying further we can
check whether features across all pairs align with our expectations. In
SSC3-23, the overall efficacies for putative excitatory connections ($n=128$)
and putative inhibitory connections ($n=85$) are comparable (median of 4$\%$
[2$\%$, 7$\%$] for excitatory, -4$\%$ [-2$\%$, -5$\%$] for inhibitory, with
intervals denoting quartiles). However putative excitatory connections tend to
occur between neurons whose distance on the electrode array is further apart
(median 308 $\mu m$ [168, 581] quartiles for excitatory and 168 $\mu m$ [119,
338] for inhibitory, Wilcoxon rank-sum $p<10^{-4}$). Time constants for
putative excitatory connections tend to be shorter (median 0.4 ms [0.3, 0.6]
quartiles for excitatory and 0.6 ms [0.5, 0.9] for inhibitory, rank-sum
$p<10^{-6}$), and latency tends to be longer (median 1.7 ms [0.2, 2.2] for
excitatory and 1.3 ms [1.1, 1.5] for inhibitory, rank-sum $p<10^{-4}$). We
also find that connection probability decays with distance (Fig 3C, inset),
and the length scale for putative excitatory connections is somewhat longer
(Binomial regression for observed counts to possible pairs at a given
distance, OR=0.50 [0.43, 0.57] per doubling for excitatory connections 0.4
[0.35, 0.47] per doubling for inhibitory connections, intervals denote 95$\%$
CI). All these observations are consistent with intracellular observations of
intracortical synaptic effects.

To illustrate some additional ways that circumstantial evidence can inform the
interpretation of individual correlograms, we switch our focus to
ABI-715093703 – a large-scale, multi-region recording with six Neuropixels
arrays from an awake, head-fixed mouse viewing a varying visual stimulus. Here
the detection step identifies 201 pairs out of 551K possible pairs (0.04$\%$)
as having statistically significant fast correlations. We again fit
correlograms for candidate pairs with a model of putative synaptic effects
(median pseudo-R2 is 0.79) and restrict analyses to more specific parameter
ranges (criteria on latency, time constant, and pseudo-$R^{2}$ as above,
51$\%$ of candidates passing, 0.02$\%$ of all possible pairs). Here we find
that the efficacies for putative excitatory connections ($n=63$) are somewhat
stronger than those for putative inhibitory connections (n=39) (median of
4$\%$ [2$\%$, 10$\%$] for excitatory, -1$\%$ [-0.6$\%$, -1.8$\%$] for
inhibitory). Time constants for putative excitatory connections tend to be
shorter (median 0.58 ms [0.37, 0.78] for excitatory and 0.74 ms [0.43, 1.08]
for inhibitory, rank-sum $p=0.06$), and latency tends to be longer (median 1.0
ms [0.8-1.5] for excitatory and 0.9 ms [0.8-1.1] for inhibitory, rank-sum
$p=0.04$).

As with SSC3-23, we can ask whether putative synaptic effects are stable. In
this case, since these recordings are from awake, behaving mice, we assess
whether they are stable across stimuli and behavior. We again find that, in
many examples, latency, duration, and sign (excitatory or inhibitory) all tend
to be reliable while efficacy and baseline postsynaptic firing rate vary (Fig
4A). Since these recordings sample multiple brain regions and multiple arrays,
we also quantify differences in putative synaptic connectivity across anatomy
(Fig 4B). Most putative synapses (88$\%$) come from neurons recorded from the
same Neuropixels probe and brain area (84$\%$). The average distance between
putative excitatory pairs on the same probe is slightly longer (84 ± 59 $\mu
m$, S.D.) than the distance between putative inhibitory pairs (72 ± 48 $\mu
m$), although not statistically significant (unpaired t-test $t(87.6)=1.1$,
$p=0.29$). Of the inter-probe ($n=12$) and inter-area connections ($n=16$),
all connections are putative excitatory connections. Inter-area connections
could also be compared to anatomical/tracing data on connectivity [54] to
verify if such connections genuinely occur or to strengthen/weaken our belief
in these specific putative synapses

Figure 5: Presynaptic dynamics can have a substantial influence on the
observed cross-correlation. (A-C) show three of the most extreme examples
taken from three recordings in the Allen Brain Observatory Neuropixels
dataset. The auto-correlations of the presynaptic (top) and postsynaptic
(bottom) neurons are shown at left, followed by the cross-correlation from all
spikes (gray). We then compute cross-correlations separately for presynaptic
spikes that were closely followed or preceded by another presynaptic spike
(Non-Isolated) and presynaptic spikes that were Isolated from nearby
presynaptic spikes by comparison (split by the median nearest spike time). (A)
and (B) illustrate cases where bursting in the presynaptic neuron causes side-
peaks in the cross-correlation (black triangles) in addition to the primary
response (red triangle). These side-peaks are less pronounced for Isolated
presynaptic spikes and can be asymmetric (filled vs open triangles in A). (C)
Illustrates a case where the shape of the presynaptic autocorrelation causes a
peak near an interval of 0, reflecting relief from inhibition (black
triangle). In all cases the red curves denote the model fit, where a single
synaptic effect is convolved with the presynaptic auto-correlation and acts in
addition to a slowly fluctuating baseline (gray). Curves for Non-Isolated and
Isolated spikes use the same synaptic model but with a case-specific shift in
the baseline constant. Note that, in the auto-correlations, the peak at
$\tau=0$ has been removed for clarity.

For this case study, we also want to demonstrate how spike waveform can play a
role as circumstantial evidence. As many past studies have noted, spike
waveforms are often clustered. Similar to previous work [11], here we find
that spike waveforms form a pair of clusters and that the waveform-type (broad
vs narrow) generally corresponds to the sign of the putative synaptic effect
(Fig 4C). There are, however, some inconsistencies (Fig 4D). One reason these
could occur may be if waveform does not uniquely identify cell type. Since
narrow waveforms can occur when an electrode is close to an axon collateral
[10], and broad waveforms can occur for inhibitory neurons [68], particularly
those that are “regularly spiking”, this is one possibility (e.g. Fig 4D,
bottom). On the other hand, the synaptic effect may not always be accurately
described despite high pseudo-$R^{2}$. With the model used here, the presence
of fast, near-synchronous correlation, in particular, can lead to
misestimation (Fig 4D, top and middle). Due to fact that the presynaptic
neurons in these examples have narrow waveforms and the fact that the peak
lacks a clear synaptic latency, these examples are unlikely to be excitatory
synapses. It is possible that these peaks are instead due to confounding
common input (see below). Altogether, these case studies demonstrate how
putative synaptic effects can be evaluated based not just on the overall
correlogram, but also based on the stability of the correlogram, anatomical
locations, spike waveforms, and the features of the population.

###  2.4 Explanatory models, presynaptic spike auto-correlation, and short-
term plasticity

In evaluating putative synapses, we particularly want to highlight the value
of explanatory models of synaptic effects. That is, models that 1) take
presynaptic spike times as input and generate predictions about the
postsynaptic spiking as output and 2) attempt to account for specific
neurophysiological mechanisms. Within this class, modelers choose whether to
predict detailed postsynaptic spike timing or to predict summary statistics,
such as correlograms, and, additionally, choose whether to describe neuron
dynamics using either membrane potentials or firing rates [18]. Here we focus
on rate-based models of the correlogram, but other descriptions may be
necessary in more complex settings [56, 77, 120]. The model fits shown above
are based on a decomposition of the correlogram into a slow, baseline
component and a fast, synaptic effect. However, to account for the dynamics of
the putative presynaptic neuron, in some cases it may be necessary to
incorporate the presynaptic autocorrelation into the model.

Due to the bursting or refractoriness of the presynaptic neuron, the influence
of the synaptic effect on the correlogram may be spread across a wide range of
intervals (Fig 5). A strong putative excitatory synapse from a presynaptic
neuron that bursts, may have multiple peaks in the correlogram corresponding
to the modes for bursts in the ISI distribution (Fig 5A and B). Similarly, a
strong putative inhibitory synapse from a presynaptic neuron with
refractoriness, may show a “release from inhibition” for intervals shortly
before 0 (Fig 5C). Many putative synapses (see examples in Fig 1, 3, and 4)
appear on a relatively constant background firing, but these cases (taken from
the Allen Brain Institute – Visual Coding Neuropixels dataset) illustrate how
presynaptic bursting and refractoriness can predictably influence the observed
correlogram.

To account for this variation, here we use a convolutional model of the fast
synaptic effect where a fast, fixed function (in this case, an alpha function)
is convolved with the presynaptic autocorrelation (see Methods). Previous
studies have noted the importance of the presynaptic autocorrelation in
shaping the cross-correlogram [16, 69] and shown how deconvolution could be
used to correct for these effects during detection, testing, and estimation
[26, 97]. Here we find that the same convolutional synaptic model (with a
single estimated latency, time constant, and strength) can describe the cross-
correlograms associated with subsets of presynaptic spikes. Here we consider
isolated presynaptic spikes (those where the preceding/following presynaptic
spikes are somewhat distant) and non-isolated presynaptic spikes (those with a
nearby preceding/following presynaptic spike), and find that, after applying
the convolution with the presynaptic auto-correlogram, a single synaptic model
describes the overall correlogram, as well as the correlograms associated with
isolated and non-isolated spikes (Fig 5).

Figure 6: Example putative synapses (Allen Institute Neuropixels) with
parameters estimated using pairwise model-based methods. A) Putative
excitatory CA1-CA1 connection (Session ID 719161530) with short-term synaptic
depression. The cross-correlogram of postsynaptic spikes (gray) shows a
transient, short latency peak following presynaptic spiking. The red curve
denotes the model fit, and black curve denotes the baseline (without the
synaptic effect). As the inter-spike interval preceding a presynaptic spike
increases the efficacy increases, suggesting recovery from depression (left).
The correlograms corresponding to presynaptic spikes short, medium, and long
ISIs are shown in blue. B) Putative excitatory VISam-VISam connection
(819701982) with short-term synaptic facilitation. C) Putative inhibitory VPM-
VPM connection (719161530) with short-term synaptic depression.

This approach accounts for some types of variation in the cross-correlation,
but it ignores the potential for short-term synaptic dynamics. The
postsynaptic response to the first spike in a burst is not expected to be the
same as the last spike in a burst due to short-term synaptic depression and
facilitation [105, 114]. Putative synapses can show patterns consistent with
short-term synaptic depression, where synaptic effects for presynaptic spikes
preceded by long ISIs are stronger than those for presynaptic spike preceded
by short ISIs (Fig 6A), or consistent with short-term synaptic facilitation,
where the effects are reversed (Fig 6B). Explanatory models of short-term
synaptic plasticity that describe resource depletion and recovery as well as
release probability and membrane potential integration can accurately describe
these patterns [28], and here we show several examples from the ABI dataset
(Fig 6, see Methods for model details). The more detailed responses to
presynaptic dynamics allow explanatory models to be tested and compared.
Models fit to data from one stimulus or brain state can be tested to see if
they are able to generalize and predict effects during other stimuli or brain
states.

###  2.5 Confounding by polysynaptic connections and common drive

As previous authors have noted, polysynaptic connections and common drive can,
in principle, confound the identification of monosynaptic connections [15, 34,
49, 71]. Strong disynaptic effects through a hidden neuron (e.g. A→H→B) may
lead to peaks or troughs in the cross-correlogram with short latencies and
durations (between neurons A and B, Fig 7A), and strong common drive (H→A and
H→B) can lead to similar synapse-like statistics when delays are coordinated
(again, between neurons A and B, Fig 7B).

To understand how common this confounding is likely to be, we consider some
approximate calculations of how efficacious polysynaptic transmission and
common drive are in practice. On the one hand, there is reason to think the
confounding may not be severe. For an excitatory disynaptic connection A→H→B
with efficacies $e_{A}H>0$ and $e_{H}B>0$, the polysynaptic efficacy of A to C
will be $\sim e_{A}H$ $e_{H}B$ assuming spike transmission is independent.
Similarly, for the case where A and B receive common drive from H, the excess
probability of A and B both spiking will be $\sim e_{H}A$ $e_{H}B$. For the
range of monosynaptic efficacies observed in vivo on the order of $\sim 5\%$,
disynaptic transmission or common drive would have an efficacy $\sim 0.25\%$.
Additionally, only a subset of confounding effects will have latencies and
durations likely to be confused for monosynaptic connections. The latencies
and durations of polysynaptic effects grow linearly with additional
connections, while the efficacy drops geometrically (Fig 6A). On the other
hand, common drive must occur with specific delays to produce latencies
consistent with monosynaptic transmission (Fig 7B). Multiple paths from A→B
(Fig 7C) or multiple hidden inputs (Fig 7D) could act to increase the
effective efficacy. However, to be comparable to a typical monosynaptic
connection many such paths would be necessary, and these paths/inputs would
need to have precisely coordinated latencies to prevent broadening of the
cross-correlation. Imprecise mixtures of polysynaptic and common drive effects
(Fig 7E) will generally not result in cross-correlograms that would be
confused for a fast monosynaptic effect.

Cross-correlations between pairs of neurons have been observed and described
on multiple timescales, and previous work has aimed to distinguish between
common input, stimulus-driven, and emergent synchrony [72, 113]. Sensory-
driven and spontaneous activity in sensory cortex can also occur with near-
synchronous spiking [8, 53, 104, 108], and similar fast correlations have been
observed between hippocampal [25] and thalamic neurons [4]. In most cases,
synchronous common input does not have a clear directionality and is centered
at zero latency (see examples from ABI-715093703 in Fig 4D and S1), but short
duration cross-correlations with non-zero latency can still have ambiguous
origins (e.g. [7]). Disynaptic effects have also been studied directly in the
projections from retina to visual cortex via the LGN [47]. Here Reid and
colleagues found that cross-correlations for disynaptic effects resemble the
convolution of retinogeniculate and geniculocortical synaptic effects [82].
The duration of the disynaptic effect is longer and the efficacy is smaller
than for each of the individual synapses, but it may be fast/strong enough
that, without accounting for anatomy, the disynaptic effect could be mistaken
for a monosynaptic effect.

Altogether, these observations suggest how polysynaptic effects and common
drive are at the same time 1) relatively unlikely to be mistaken for
monosynaptic effects, on average, and 2) possible confounds in concrete cases.
The short duration and non-zero latency of monosynaptic transmission provides
some protection against confounding by slow effects, but the overall cross-
correlogram alone cannot uniquely differentiate between monosynaptic effects
and other fast dependencies between pre- and postsynaptic spikes. The
circumstantial evidence discussed above, however, may provide an additional
layer of defense against mistaking polysynaptic effects and common drive for
monosynaptic transmission. In cases where cross-correlations have a short-
latency, transient peak or trough due to confounding, other evidence may argue
against the presence of a monosynaptic connection: Dale’s Law may be violated,
spike waveforms may be inconsistent with observed “excitation” or
“inhibition”, or effects may be improbable given the anatomical locations of
the putative pre- and postsynaptic neurons.

Figure 7: Spurious correlations can occur due to polysynaptic chains and/or
common input. However, in order to be mistaken for monosynaptic effects, these
confounders need to have specific structure. A) Since the efficacy of single
synapses tends to be much less than 1, the strength of single polysynaptic
chains decays rapidly. The latency and time constant of the effect also grows
with the number of connections. Thus, only chains with strong connections and
relatively few links are likely to be mistaken for direct, monosynaptic
connections. B) Common drive from hidden neurons can also create spurious
correlations. Again, since the efficacy of single connections is much less
than 1, these effects will tend to be weaker and somewhat less temporally
precise. Additionally, unless the latencies of the connections are
substantially different, the effect of common drive will be centered at zero
delay and, thus, be less likely to be mistaken for a monosynaptic connection.
Although the effects of individual polysynaptic chains or common drive from an
individual input may be weak, multiple chains (C), multiple drivers (D), or a
combination of the two (E) can create spurious correlations comparable in
strength and timing to monosynaptic connections, albeit with even less
temporal precision than single inputs/chains. Figure 8: Examples of putative
subnetworks involving three single units. A) Auto-correlograms and cross-
correlograms for putative common drive from a single unit with a broad
waveform to two units with narrow waveforms (VIS, Session ID 744228101).
Triangle denotes the potential, spurious correlation between the two narrow
waveform units due to their shared excitatory input. B) Putative feed-forward
inhibition (VISal, 760693773), C) Putative inhibitory subnetwork (VISrl,
746083955). Triangles denote the synchronous activity of all three units. D)
Putative microcircuit with a reciprocal connection (grey, 742951821). Triangle
denotes the putative reciprocal inhibition. Correlogram columns within each
panel correspond to the same putative postsynaptic neurons and have the same
axes. Putative presynaptic units whose spike waveforms were classified as
broad are colored in red, while those classified as narrow are shown in blue.

###  2.6 From putative synapses to putative microcircuits

Careful consideration of confounding is necessary whenever there are
unobserved influences on postsynaptic spiking [70], but, as the scale of
multi-electrode recordings increases, observed disynaptic effects (e.g. A→B→C)
and observed diverging outputs from a common presynaptic neuron (e.g. A→B and
A→ C) may also be studied more directly, alongside other putative microcircuit
motifs. In the ABI datasets, there are many examples of neurons that appear to
be, not just synaptically connected as an isolated pair, but connected in
larger subnetworks (Fig 8). In one example (Fig 8A), a putative excitatory
presynaptic neuron (with a broad waveform) appears to project to two putative
inhibitory neurons (with narrow waveforms). The weak, near-synchronous
correlation between the two inhibitory neurons may be due to their common
input. More complex patterns are also common – for instance, putative feed-
forward inhibition (Fig 8B), putative inhibitory subnetworks (Fig 8C), and
putative networks with reciprocal connections (Fig 8D).

Motifs are hypothesized to be building blocks of information processing [13,
118], and there are widespread similarities in microcircuits across species
and brain regions [94]. Although motifs have been observed in previous studies
of putative synapses [11, 31], description and modeling of these motifs during
behavior is still in the early stages. Despite the unmodeled influence of
unobserved inputs, explanatory models allow direct predictions of the motif
effects. Feed-forward inhibition in the cerebellum, for instance, has been
well-modeled by the combination of putative synaptic effects [12]. As large-
scale spike recordings increase in scale, searching for and modeling putative
synaptic effects may complement intracellular studies of these motifs e.g.
[79, 119] and allow predictions based on in vitro studies to be tested in
behaving animals.

##  3 Discussion

Here we have argued that synaptic connections can reasonably be studied from
large-scale spike recordings. As the scale of spike recordings increases these
connections will inevitably occur in the data, and there is strong evidence
that at least some synapses can be detected from spikes alone under realistic
physiological conditions. We show how putative connections detected during
observation are often supported by multiple lines of circumstantial evidence,
and, although confounding is still a concern, models built with synaptic
effects can provide compelling explanations for the detailed structure of
pairwise spike statistics. These models do not give unbiased estimates of
verified, causal synaptic effects when fit to observational data, but they do
provide testable descriptions of the data with synaptic parameters, dynamics,
or microcircuit structure.

Here we have particularly emphasized the utility of detailed explanatory
models and intersecting evidence from sources other than the overall
correlogram for interpreting putative synapses. Although, the overall cross-
correlogram between the spikes of neuron A and neuron B provides “merely”
observational evidence for the potential presence of a synapse, our confidence
in the presence of a synapse can be strengthened or weakened by integrating
prior physiological knowledge and by testing whether explanatory models make
accurate predictions. We have focused here largely on the qualitative role
that circumstantial evidence can play in interpreting spike correlations, but,
in some cases, it may be useful to quantitatively integrate circumstantial
evidence into the detection and modeling process, for instance, using Bayesian
methods [84]. Here we have reviewed how information about network structure
(e.g. Dale’s law), anatomy, and spike waveforms can be used to assess whether
candidate correlograms are consistent with synaptic effects. And we have shown
how detailed consideration of the latency and duration of synaptic effects as
well as modeling of presynaptic dynamics can further alter our confidence. In
observational studies, this “multi-phasic” approach to evaluating the evidence
[21] has been called “Cochran’s crossword” [86]. Although we may be uncertain
about a specific answer (the presence of a synapse) given a single clue (the
overall cross-correlogram), intersecting clues can, like a crossword puzzle,
support or contradict our belief that two neurons may be connected.

Many methods have been developed for measuring statistical dependencies
between pairs of neurons [1]. Although the correlogram itself, and other
measures such as Granger causality and transfer entropy, can provide important
descriptions of dependencies between pairs of spike trains, the scientific
goal of the explanatory models described here is somewhat distinct. Having a
detailed explanatory model [39, 91] that predicts postsynaptic responses
allows measurement but also allows multiple explanations to be compared and
falsified. Comparisons beyond the data, for instance, to anatomy [32] or
across stimuli [36], play an essential role in the interpretation of the
results. New models that incorporate either more biophysical detail, such as
membrane dynamics [56, 77], or more complex forms of spike transmission and
plasticity [63, 95, 120] are actively being developed, and these models may
provide even more accurate descriptions and predictions of postsynaptic
spiking. Accurate descriptions of the fluctuations in postsynaptic
excitability and adaptation (c.f. [52]) may be especially important, since
these effects often occur on similar timescales to short- and long-term
plasticity.

When the standard of evidence is strict, errors and biases in the
identification of putative synapses are not qualitatively worse than errors
and biases in spike sorting [116], firing rate estimation [5], cell type
identification [57, 109], sampling of units, or the targeting of brain
regions. However, for studying synaptic effects, one major bias is in the
strength of detected connections – strong synapses between neurons with high
firing rates are more likely to be detected than weak connections between
neurons with low firing rates. False positives may occur if common input is
mistaken for a synaptic effect, but there are likely to be many, many false
negatives. The synapses that are detected, sometimes referred to as the
“billionaires” (Swadlow, personal communication), are likely to be
unrepresentative of typical synaptic effects and many not accurately reflect
typical microcircuit operation. Even if we can be confident that a putative
synapse is likely to be a genuine synapse, it is important to keep in mind
that we may be studying the most impactful synapses and not the typical
synapses. Here we have focused exclusively on efficacy as a measure of impact.
However, many previous studies emphasize contribution rather than or in
addition to efficacy (c.f. [67]). Note that for excitatory synapses,
contribution can be rather directly calculated by scaling the efficacy by
$N_{pre}/N_{post}$ where $N$ denotes total spikes for the pre- or postsynaptic
neuron. We focus on efficacy here, since contribution is ill-defined for
inhibitory synapses.

Without experimental interventions, there will always be uncertainty about
whether a specific pair of neurons is genuinely synaptically connected, and
unobserved confounds can generally lead to biases in the estimated
dependencies between neurons, just as they do for other estimates [98].
Nonetheless, here we argue that the problem is, perhaps, not as dire, as
several recent results have suggested [23, 61, 66]. In the case of putative
synapses, circumstantial evidence and explanatory models can act to mitigate
concerns about confounding, and focusing on these effects allows synaptic
transmission to be studied in intact neural systems. However, efforts to
identify putative synapses should be distinguished from other efforts to
describe functional connectivity [81, 100] that are at least an order of
magnitude slower and denser.

##  4 Methods

Code for the results shown here is available at
https://github.com/stevensonlab/syngalong$\\_$v0-1

###  4.1 Simplified power analysis

Consider a hypothesis test of a synaptic connection with efficacy $e$ given
$N$ presynaptic spikes and known probability $p$ for the postsynaptic neuron
firing by chance within a fixed detection window following each presynaptic
spike. We wish to decide between two hypotheses: $H_{0}:e=0$ versus
$H_{A}:e\neq 0$. Assuming that the observations are independent, the null
distribution for the number of postsynaptic spikes in the detection window is
given by $y\sim Binomial(N,p)$ and alternative distribution is $y\sim
Binomial(N,p+e)$. Using the normal approximation to the binomial distribution
$Binomial(N,p)\approx Normal(Np,\sqrt{Np(1-p)}$, with confidence level
$\alpha$, the approximate power $(1-\beta)$ is given by

$1-\beta=1+\Phi\left(\frac{\sqrt{Ne^{2}}+\sqrt{p(1-p)})z_{\alpha/2}}{\sqrt{(p+e)(1-p-e)}}\right)-\Phi\left(\frac{\sqrt{Ne^{2}}+\sqrt{p(1-p)}z_{1-\alpha/2}}{\sqrt{(p+e)(1-p-e)}}\right)$

where $\Phi(\cdot)$ denotes the CDF for the standard normal distribution and
$z_{\cdot}$ denotes the quantiles of the standard normal distribution (e.g.
$z_{0}.975=1.96$). We can approximately solve for the efficacy required to
achieve a given power using a first-order Taylor expansion around e=0. We have
two solutions, one for excitatory connections and one for inhibitory

$\displaystyle e^{+}$
$\displaystyle\approx\frac{p(1-p)\left(z_{1-\alpha/2}-z_{\beta}\right)}{\sqrt{Np(1-p)}-(p-\frac{1}{2})z_{\beta}}$
$\displaystyle e^{-}$
$\displaystyle\approx\frac{p(1-p)\left(z_{\alpha/2}-z_{1-\beta}\right)}{\sqrt{Np(1-p)})-(p-\frac{1}{2})z_{1-\beta}}$

These are upper-bounded and equal when $p=0.5$, in which case
$e_{p=0.5}\approx(z_{1-\alpha/2}-z_{\beta})/(2\sqrt{N})$. For the commonly
used values of $\alpha=0.05$ and $1-\beta=0.8$ this gives $e_{p=0.5}\approx
1.4/\sqrt{N}$.

In practice, there are major caveats to this analysis: 1) it ignores the
problem of how to define the time window of interest, 2) it ignores the fact
that the shape/level of the baseline p is unknown, 3) it ignores the
dependencies that exist between post-synaptic spikes, and 4) it ignores the
complex, overdetermined relationships between biophysical parameters and
efficacy. Many methods for detecting synapses from spikes have aimed to
address Caveats 1, 2, and 3. However, with these computational and model-based
methods, simulations are necessary to provide estimates of detection accuracy
in specific settings. Rather than an exact power analysis, these results often
focus on generating estimates of false positive rates ($\hat{\alpha}$) and
false negative rates ($\hat{\beta}$). Caveat 4, is an important consideration
for linking results with extracellular recordings to the many previous
intracellular studies of synaptic transmission [56, 77]. Efficacy cannot be
directly inferred from post-synaptic currents or potentials (PSC/PSPs) but
depends on the nonlinear transformation between membrane potential and spikes.
Differences in receptor time constants (e.g. AMPA vs NMDA or GABAA vs GABAB)
and the presence of additional currents, mean that two synapses with similar
PSC/PSPs may have quite different efficacies and two synapses with the same
efficacies can have quite different biophysical origins. Considering efficacy
directly prevents us from making precise conclusions about synaptic biophysics
but allows us to derive approximate limits for detecting synaptic connections.

###  4.2 Data

Here we examine two datasets as case studies: 1) an in vitro recording of
spontaneous activity from an organotypic slice culture of somatosensory
cortex, and 2) an in vivo recording from multiple Neuropixels arrays in an
awake mouse.

Slice data were previously collected and are described in detail in [46].
Briefly, recordings were made using a large-scale multielectrode array (1 x 2
mm total area with 512 electrodes spaced 60 $\mu m$ apart on a hexagonal
lattice). Single units were visually identified and sorted using multichannel
waveforms from each electrode and its six neighbors. Data are available via
the Collaborative Research in Computational Neuroscience (CRNCS) Data Sharing
Initiative: http://dx.doi.org/10.6080/K07D2S2F.

Neuropixels recordings from the Allen Institute for Brain Science – Visual
Coding dataset were previously collected and described in detail in [93].
Briefly, head-fixed mice viewed a standardized set of visual stimuli
(including Gabor patches, full-field drifting gratings, moving dots, and
natural images and movies) while they were free to run on a wheel. Data are
available from the Allen Brain Institute (https://portal.brain-
map.org/explore/circuits).

###  4.3 Detecting putative monosynaptic connections

We detect putative connections based on the cross-correlograms between pairs
of neurons. Several approaches for detecting putative synapses have been
developed based on hypothesis testing [6, 31] or model comparison [51, 84].
Here, since we are focused on the modeling and interpretation of putative
synapses rather than detection per se, we detect putative monosynaptic
connections using a computationally efficient, approximate hypothesis test
based on the jitter method. Namely, we compare the observed cross-correlogram
to a version of the cross-correlogram that has been smoothed and evaluate the
probability of the observed spike count in each bin under a null distribution.

For the exact jitter method, a null distribution for $c(\tau)$ is generated
based on the empirical correlograms when observed spike times are jittered.
Here we evaluate p-values $c(\tau)$ based on a null distribution
$Binomial(s(\tau),p(\tau))$ that assumes individual counts in the correlogram
are conditionally independent given $\tau$. The trials $s$ and probability $p$
associated with each bin of the correlogram are generated by a “transport”
function $g(t)$. Conceptually, $s(\tau)$ reflects the number of spikes that
could be jittered from neighboring bins into bin $\tau$

$s(\tau)=\sum_{t}\mathbb{1}_{|\tau-t|<m}c(\tau-t)g(t)$

and $p(\tau)$ reflects the probability of a spike being transported into the
given bin

$p(\tau)=1/\sum_{t}\mathbb{1}_{|\tau-t|<m}g(t)$

Here we use a Gaussian window $g(t)=exp(-\frac{t^{2}}{2\sigma^{2}})$ with a
standard deviation of $\sigma=2ms$. Note that when $\sigma$ is bigger than the
binsize, $\sum_{t}g(t)>1$ and $s(\tau)>c(\tau)$. More spikes could land in bin
$\tau$ than actually do. When the correlogram is calculated over a fixed,
symmetric interval $-m\geq\tau\geq m$ and $\sigma<<m$, $p$ is approximately
constant but increases to account for the edges of the correlogram. Since this
approach is based on simply smoothing the correlogram, it is substantially
faster than the simulation-based jitter method, and effectively detects any
variation in the correlogram that is fast compared to the timescale of the
smoothing.

When there are strong dependencies between individual counts in different bins
this approximation will not be as accurate as the jitter method. For example,
if the pre-synaptic neuron bursts nearby post-synaptic spikes will appear as
non-independent counts in the correlogram. However, in practice, we find that
the independence assumption detects reasonable connections with substantial
computational savings.

Testing B bins within $1ms<\tau<25ms$ of each correlogram with C neurons
results in $O(BC^{2})$ hypothesis tests. To correct for multiple comparisons,
we use a False Discovery Rate method (Benjamini-Hochberg procedure) with
$\alpha=0.1$ and consider pairs with at least two statistically significant
time bins to be a putative connection. Increasing $\alpha$ would result in
more pairs being detected with greater numbers of weaker connections and more
false positives [84].

In some cases, overlapping spike waveforms from nearby neurons (spike
shadowing) leads to misestimation of $c(\tau)$ near $\tau=0$ [76, 80]. Here we
detect these cases using a spike shadowing index

$SSI=min\left(\frac{\left|\bar{c}_{|\tau|<t_{w}}-\bar{c}_{-2t_{w}<\tau<-t_{w}}\right|}{\bar{c}_{-2t_{w}<\tau<-t_{w}}},\frac{\left|\bar{c}_{|\tau|<t_{w}}-\bar{c}_{t_{w}<\tau<2t_{w}}\right|}{\bar{c}_{t_{w}<\tau<2t_{w}}}\right)$

This index compares the average observed count (denoted $\bar{\cdot}$) in the
window of potential overlap $|\tau|<t_{w}$ to the neighboring regions of the
correlogram on either side and is large when there is abrupt variation in the
correlogram near $\tau=0$. Here we set $t_{w}=1ms$ and exclude pairs of
neurons with $SSI>0.4$ (7$\%$ of pairs, in Fig. 1). Recent spike sorting
methods that explicitly consider overlapping waveforms [73] typically have
much less spike shadowing and may correct for sorting errors due to changing
spike shape. For putative synapses between units on different arrays or on
electrodes that are far apart, spike shadowing is not a problem and $SSI$ is
often near 0.

###  4.4 Modeling synaptic effects in the cross-correlogram

Although the hypothesis-test based approach is effective for rapidly detecting
connections, it only looks for differences between fast and slow structure in
the correlogram and does not provide a generative model for synaptic
transmission. In order to assess whether putative connections are excitatory
or inhibitory, measure their latencies, and efficacies we fit an extended GLM,
similar to [51, 84].

Namely, we model the counts in the cross-correlogram using Binomial regression
with

$p_{\tau}=f\left(\beta_{0}+X_{\tau}\beta+a_{\tau}*w\alpha(t)\right)$

where $X_{\tau}$ denotes a set of fixed basis functions weighted by
coefficients $\beta$ to describe slow fluctuations in the baseline
correlogram, while the fast synaptic effect is modeled by convolving an alpha
function
$\alpha(t)=\frac{t-t_{0}}{\tau_{\alpha}}exp\left(-\frac{t-t_{0}}{\tau_{\alpha}}\right)$
with the auto-correlation of the presynaptic neuron $a_{\tau}$ with synaptic
weight $w$.

Here $t_{0}$ and $\tau_{\alpha}$ denote the latency and time constant for the
synaptic effect, respectively, and the convolution allows temporal
dependencies in the presynaptic firing (e.g. burstiness) to be taken into
account. Here, to model the slow fluctuations in the correlogram we use a set
of cubic B-spline basis functions with equally spaced knots (between 3-6 for
the fits shown here). The combination of the slow effect and fast synaptic
effect is then passed through a logistic nonlinearity $f(x)=1/(1+e^{-x})$, and
the number of counts observed in the correlogram is assumed to be distributed
following $Binomial(N,p_{\tau})$ where $N$ is the number of presynaptic
spikes.

The parameters $\theta=\left\\{\beta_{0},\beta,w,t_{0},\tau_{\alpha}\right\\}$
are optimized by maximizing a penalized log-likelihood

$logp(c|\theta)\propto\sum_{\tau}\left[c_{\tau}log(p_{\tau})+(N-c_{\tau})log(1-p_{\tau})\right]-\eta_{\beta}||\beta||^{2}-\eta_{w}w^{2}-\eta_{\tau}\tau_{\alpha}$

using gradient descent, and the hyperparameters
$\eta=\left\\{\eta_{\beta},\eta_{w},\eta_{\tau}\right\\}$ act to regularize
the basis function coefficients, synaptic amplitude, and synaptic time
constant, respectively. Due to the parameterization of the alpha function, the
log likelihood is not concave, and we use random restarts to find the best
solution. The synaptic latency and time constant are both constrained to be
positive by log-transforming during optimization. After optimization, the
efficacy is estimated by

$e=\sum_{\tau\in\alpha>0.01}(p_{\tau}-f(\beta_{0}+X_{\tau}beta))$

To account for asymmetry in the effect of the auto-correlogram related
responses we, in some cases (e.g. Fig 5A), separate the auto-correlogram into
two segments and add the additional parameter $v$

$p_{\tau}=f\left(\beta_{0}+X_{\tau}\beta+a_{\tau<0}*v\alpha(t)+a_{\tau\geq
0}*w\alpha(t)\right)$

The log-likelihood itself can be used for detecting putative synapses [84],
but here we focus on the model fits of already suspected synapses, using
pseudo-$R^{2}$ to assess goodness-of-fit

$R^{2}=\frac{L_{\theta}-L_{0}}{L_{s}-L_{0}}$

where $L_{\theta}$ denotes the (unpenalized) log-likelihood for the fit model,
$L_{0}$ the log-likelihood for the null model (with constant $p=\bar{c}/N$),
and $L_{s}$ the log-likelihood for the saturated model (with
$p(\tau)=c(\tau)$).

###  4.5 Modeling short-term synaptic plasticity

Following [28] we fit a nonlinear model for the efficacy as a function of the
presynaptic inter-spike interval $\Delta t$…

$\displaystyle S(\Delta t)$ $\displaystyle=Uexp\left(-\frac{\Delta
t}{\tau_{s}}\right)$ $\displaystyle D(\Delta t)$
$\displaystyle=1-Uexp\left(-\frac{\Delta t}{\tau_{d}}\right)$ $\displaystyle
F(\Delta t)$ $\displaystyle=U+\left[U(1-U)\right]exp\left(-\frac{\Delta
t}{\tau_{f}}\right)$ $\displaystyle\hat{e}(\Delta t)$
$\displaystyle=A\left[S(\Delta t)+F(\Delta t)D(\Delta t)\right]$

Where $S(\cdot)$, $D(\cdot)$, $F(\cdot)$ denote effects for synaptic
integration, short-term synaptic depression, and short-term synaptic
facilitation, respectively. $A$ denotes the amplitude of the synaptic effect,
$U$ denotes the release probability, and $\tau_{\cdot}$ denotes the time
constants of the three effects. The model here corresponds to the predictions
of the Tsodyks-Markram model for a pair of presynaptic spikes that starts from
a fully recovered state [111]. We fit the parameters
$\theta=\left\\{A,U,\tau_{s},\tau_{d},\tau_{f}\right\\}$ by minimizing the sum
squared error $\sum_{\Delta t}\left(e(\Delta t)-\hat{e}(\Delta t)\right)^{2}$
between the model predicted ISI-dependency $\hat{e}(\Delta t)$ and the
observed ISI-dependency $e(\Delta t)$. Here $e(\Delta t)$ is estimated by
fitting the correlograms of subsets of presynaptic spikes that have specific
quantiles of presynaptic ISIs $\Delta t$. We use the model above to first
estimate the synaptic latency $t_{0}$ and time constant $\tau_{\alpha}$ for
the overall correlogram, but reoptimize $w$ based on the subsets of
presynaptic and their corresponding autocorrelograms. For the short-term
plasticity model we use random restarts to avoid local minima, and, during
optimization, the parameters are transformed
$\theta^{\prime}=\left\\{A,log(\frac{U}{1-U}),log(\tau_{s}),log(\tau_{d}),log(\tau_{f})\right\\}$
to constrain $0<U<1$ and $\tau_{\cdot}>0$.

## acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant 1651396. Thanks to Harvey Swadlow, Monty Escabí, Abhijith Mankili,
Abed Ghanbari, and Naixin Ren for helpful discussions. Thanks to Shinya Ito,
Alan Litke, and John Beggs and to the Allen Institute for Brain Science for
sharing their datasets and for supporting open science.

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Figure S1: Correlograms for $n=102$ pairs detected from ABI-715093703 screened
for short latency and time constant and for high pseudo-$R^{2}$. Blue
correlograms denote intra-area pairs, red correlograms denote pairs between
areas. Labels correspond to the unique unit ids. 950948527>950948547, for
instance, denotes a correlogram calculated “from” unit id 950948527 “to” unit
id 950948547.