Burst-Time-Dependent Plasticity Robustly Guides ON/
OFF Segregation in the Lateral Geniculate Nucleus
Julijana Gjorgjieva1*, Taro Toyoizumi2, Stephen J. Eglen1
1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom, 2 Department of Neuroscience, Columbia
University, New York, New York, United States of America

Abstract
Spontaneous retinal activity (known as ‘‘waves’’) remodels synaptic connectivity to the lateral geniculate nucleus (LGN)
during development. Analysis of retinal waves recorded with multielectrode arrays in mouse suggested that a cue for the
segregation of functionally distinct (ON and OFF) retinal ganglion cells (RGCs) in the LGN may be a desynchronization in
their firing, where ON cells precede OFF cells by one second. Using the recorded retinal waves as input, with two different
modeling approaches we explore timing-based plasticity rules for the evolution of synaptic weights to identify key features
underlying ON/OFF segregation. First, we analytically derive a linear model for the evolution of ON and OFF weights, tounderstand how synaptic plasticity rules extract input firing properties to guide segregation. Second, we simulate
postsynaptic activity with a nonlinear integrate-and-fire model to compare findings with the linear model. We find thatspike-time-dependent plasticity, which modifies synaptic weights based on millisecond-long timing and order of pre- andpostsynaptic spikes, fails to segregate ON and OFF retinal inputs in the absence of normalization. Implementing
homeostatic mechanisms results in segregation, but only with carefully-tuned parameters. Furthermore, extending spike
integration timescales to match the second-long input correlation timescales always leads to ON segregation because ON
cells fire before OFF cells. We show that burst-time-dependent plasticity can robustly guide ON/OFF segregation in the LGN
without normalization, by integrating pre- and postsynaptic bursts irrespective of their firing order and over second-longtimescales. We predict that an LGN neuron will become ON- or OFF-responsive based on a local competition of the firing
patterns of neighboring RGCs connecting to it. Finally, we demonstrate consistency with ON/OFF segregation in ferret,despite differences in the firing properties of retinal waves. Our model suggests that diverse input statistics of retinal waves
can be robustly interpreted by a burst-based rule, which underlies retinogeniculate plasticity across different species.
Citation: Gjorgjieva J, Toyoizumi T, Eglen SJ (2009) Burst-Time-Dependent Plasticity Robustly Guides ON/OFF Segregation in the Lateral Geniculate Nucleus. PLoS
Comput Biol 5(12): e1000618. doi:10.1371/journal.pcbi.1000618
Editor: Lyle J. Graham, Université Paris Descartes, Centre National de la Recherche Scientifique, France
Received July 23, 2009; Accepted November 17, 2009; Published December 24, 2009
Copyright: ß 2009 Gjorgjieva et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was funded by a Cambridge Overseas Research Studentship and Trinity College Internal Graduate Studentship (JG), and the Robert Leet andClara Guthrie Patterson Trust Postdoctoral Fellowship, and Bank of America Trustee (TT). The funders had no role in study design, data collection and analysis,
decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: jg447@cam.ac.uk(STDP): synaptic change is induced from pairing multiple pre- andpostsynaptic spikes, firing within tens of milliseconds of each other[9,10]. Various extensions which include triplets, quadruplets andother nonlinearities in spike integration have also been studied[11–13], but they commonly predict synaptic potentiation if
presynaptic activity shortly precedes postsynaptic activity, andsynaptic depression otherwise. One may argue that in developing
systems, immature synapses are incapable of encoding information
using precisely-timed spikes, but use bursts over coarser timescales[14,15]. Butts et al. [16] recently proposed burst-time-dependent
plasticity (BTDP), based on recordings at the developing
retinogeniculate synapse, as an alternative to STDP. In BTDP,
synaptic change is induced according to the timing of bursts overlonger timescales of a second, and irrespective of the firing order of
pre- and postsynaptic bursts.
To compare spike- and burst-based mechanisms in theremodeling of synaptic connections in a realistic developmental
scenario, we examine the segregation of ON and OFF RGCs
(which respond to light increments and decrements, respectively)
onto postsynaptic neurons in the LGN. Early in development,
individual LGN neurons receive inputs from w20 mixed ON andIntroduction
During the development of the visual system, connections
between neurons form and refine in a self-organized manner
governed by various mechanisms. Initially, target neurons arecontacted by multiple RGCs following gradients of molecular cues
[1–3]. As circuits mature, visually-evoked activity maintains theseconnections; however, in early development when photoreceptors
are functionally inactive, activity is spontaneously generated within
the retina. This spontaneous activity spreads across the retina in
the form of waves, and is believed to encode different cues forsynapse maturation in the visual system [4]: as inappropriate
connections are eliminated, appropriate connections are strengthened following Hebbian-like coincidence detection mechanisms[5,6]. There is a long-standing, and still active, debate about therelative importance of activity-dependent mechanisms in development [7,8]. Theoretical models can help inform this debate byevaluating hypotheses about the role of neural activity in theremodeling of connections.
One possible mechanism of coincidence detection of pre- andpostsynaptic activity is that of spike-time-dependent plasticity
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ON/OFF Segregation in the LGN

firing patterns of the two input types, when we extend thesetimescales of STDP to half a second, segregation is always biased
towards the cell that fires first, ON. We find that BTDP canrobustly drive ON/OFF segregation in the LGN without requiringnormalization, because in addition to the second-long timescales, it
integrates pre- and postsynaptic activity irrespective of temporalorder. The segregation outcome results from a trade-off between
the shorter high-frequency bursts of ON cells, and the morestrongly correlated OFF cells. Despite differences in firing
patterns, our model can also reproduce ON/OFF segregation inferret [22], suggesting a universality of the rule which governs
ON/OFF segregation across species.

Author Summary
Many central targets in the brain are involved in theprocessing of information from the outside world. Before
information about the visual scene reaches the visualcortex, it is preprocessed in the retina and the lateralgeniculate nucleus. Connections which relay this information between the different brain targets are not determined at birth, but undergo a developmental period
during which they are guided by molecular cues to thecorrect locations, and refined by activity to the appropriate
numbers and strengths. Before the onset of vision,
spontaneous activity generated within the retina plays
an important role in the remodeling of these connections.
In a computational and theoretical model, we usedrecorded spontaneous retinal activity patterns with several
plasticity rules at the retinogeniculate synapse to identify
the key properties underlying the selective refinement ofconnections. Our model shows robust behavior whenapplied to both mouse and ferret data, demonstrating thata common plasticity rule across species may underlie
synaptic refinements in the visual system driven by
spontaneous retinal activity.

Results
In Figure 1A we illustrate the model used in this study,
consisting of multiple ON and OFF RGC inputs projecting in a
feedforward manner to a postsynaptic neuron in the LGN. Inputs
to the model were experimentally-recorded activity patterns in theform of spike trains from P12 mouse retina by Kerschensteinerand Wong [20], with 0:1 ms precision and duration of *1 hour.
Six experimental data sets were used (all recordings available
online as Dataset S1), each consisting of 5–8 mixed (ON and OFF)RGC inputs recorded by one field of the multielectrode array usedby Kerschensteiner and Wong [20]. Due to the high density ofelectrodes within one recording field (30 mm between electrodes),differences in distance-dependent correlation among RGCs within
the same data set were not relevant at this age [29]. RGCs werepreviously identified by Kerschensteiner and Wong as ON or OFF
by light modulation; at the age studied, 93% of the total recorded
RGCs showed a response exclusively to light onset or offset [20].
This suggests that relatively few RGCs at this stage display mixed
ON and OFF responses. We report input statistics (firing rates,
numbers of ON and OFF inputs in each data set) in Table 1.
Given these inputs, the activity of the postsynaptic LGN neuron
was generated using two different models (Materials and Methods):
in a biologically-realistic model we simulated the postsynaptic
neuron with nonlinear integrate-and-fire dynamics and used as
inputs the spike trains of all ON and OFF RGCs in each data set;in a reduced model we based postsynaptic activity on a linear
Poisson neuron, and derived a linear equation for the dynamics oftwo synaptic weights, one ON and one OFF, using input statistics
fits of the most correlated ON/ON and OFF/OFF cell pairs(Figure 1B). Change in synaptic strength was governed by STDP
(Figure 2A) or BTDP (Figure 2B), where Figure 2C illustrates how
bursts were detected. The two modeling approaches (linear
Poisson and nonlinear integrate-and-fire models) complement
each other in identifying which plasticity rule can read out key
retinal wave properties for ON/OFF segregation.

OFF RGCs; these inputs segregate such that eventually LGN
neurons receive inputs from 1–3 RGCs of the same type (ON or
OFF) [6,17]. Blocking spontaneous activity inhibits this segregation [18,19]. Recent experiments in mouse identified a differencein the firing patterns of RGCs that might instruct ON/OFF
segregation: at P12, cells of the same type fire together, but OFF
RGCs fire about a second after ON RGCs [20]. This asynchrony
differs from that found in ferret, where OFF cells fire more often
than ON cells [21,22].
Here, we report results from a modeling study of the propertiesof experimentally-proposed synaptic plasticity rules and modifications to these rules necessary to capture the segregation of ON and
OFF retinal inputs to a postsynaptic LGN neuron driven byrecorded RGC spike trains [20]. We take two approaches toinvestigate this problem: (i) by making various assumptions, wereduce the system to one that is analytically tractable, to allow us
to use eigenvalue theory to predict synaptic weight development[22–24]; (ii) computational simulations allow us to test a larger
system with less-restrictive assumptions. Combining these approaches gives us insight into why the models perform as they do.
Unsurprisingly, we find that a naively-implemented (additive) pairbased STDP [25] cannot segregate ON and OFF inputs using
experimental values for the balance between synaptic potentiationand depression [9,10,26]. Since STDP integrates spikes over tensof milliseconds, on this timescale ON/OFF correlations are muchsmaller than ON/ON and OFF/OFF correlations, and both ON
and OFF RGCs successfully drive the postsynaptic neuron leading
to potentiation of both cell types. Therefore, homeostaticmechanisms must be implemented to induce synaptic weightcompetition [27,28]. However, for biological ratios of depression
to potentiation close to unity, the segregation outcome is highly
sensitive to the choice of parameters. This sensitivity aries since
STDP with approximately balanced potentiation and depression
does not utilize the correlations between cells but rather the cells’
time-averaged firing rates. Within the millisecond-long integration
timescales, the second-long RGC correlations appear noisy andessentially constant. Due to the asymmetry of STDP potentiationand depression effects cancel each other, such that the signaling
component of STDP for segregation is diminished. Since longer
integration timescales are more appropriate for capturing the
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Standard STDP with 20 ms timescales for spike
integration fails to guide ON/OFF segregation in realistic
parameter regimes
First, we studied a standard pair-based STDP rule with additive
synaptic change (Figure 2A) [25] under the reduced linear model
framework (Materials and Methods) [24]. Comparing typical input
correlation timescales used by the linear model (for example, data
set 1 in Figure 1B) with the timescales of spike integration in STDP
(Figure 2A), suggested that the resulting weight dynamics would bedetermined by the relative strength of the correlation between
ON/ON and OFF/OFF pairs for a small time lag around 0 ms
(Figure 1B, top and middle panels), because the millisecond-longcorrelation window of STDP would ignore the +1 second offset
in the ON/OFF and OFF/ON correlation pairs (Figure 1B,
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ON/OFF Segregation in the LGN

Figure 1. Model description and input correlation functions. (A, left) An LGN neuron receives feedforward weak synaptic input fromneighboring ON (red) and OFF (blue) RGC inputs early in development. (A, right) Spontaneous retinal waves selectively refine RGC inputs, such thatsynaptic weights of one RGC type (ON) strengthen, while weights of the other RGC type (OFF) decay to 0, resulting in an ON-responsive LGN neuron.
Sample P12 spike rasters in the middle demonstrate that ON cells fire shorter bursts of higher spike frequency, while OFF cells fire longer bursts oflower spike frequency, *1 second after ON [20]. Also shown are spike rasters for a developing LGN neuron receiving weak mixed ON and OFF inputs,
and for a refined LGN neuron, receiving selective input solely from ON RGCs. LGN spiking activity was generated using the integrate-and-fire model inEquations 12–13. (B) Correlation functions for the input spike trains shown in (A) (mouse data set 1). The input correlation function for RGCs of thesame type (ON/ON and OFF/OFF) peaks at *0 seconds, while for RGCs of different type peaks at *{1 second for ON/OFF and *z1 second for
OFF/ON pairs. Symmetric decaying exponentials were fit to the pairwise input correlations using Equation 11. Peak amplitudes and decay time
constants for pairs with maximal peaks are reported in Table 1.
doi:10.1371/journal.pcbi.1000618.g001

integrals of STDP, A{ t{ =Az tz (Figure 2A). Most experimental
data have reported similar integration timescales for depression
and potentiation, and approximately equal amplitudes [30], or

bottom panel). We explored segregation outcome as a function of
the balance between depression and potentiation effects in theplasticity rule, denoted by the ratio of the negative and positive

Table 1. Peak amplitudes and decay time constants of the symmetric fall-off exponential fits to the correlation functions of themost-correlated input pair from spontaneous retinal wave recordings by Kerschensteiner and Wong [20] (values given as
estimate+standard error). SSON=ON T(t) and SSOFF=OFF T(t) denote the average firing rates of the two most correlated ON and OFF
cells in each data set; other parameters are illustrated in Figure 1B.

Set

1

2

3

4

5

6

# ON cells

3

6

3

5

4

3

# OFF cells

3

2

2

2

2

5

1.51

0.40

0.70

1.01

0.34

1.04

SSON=ON T(t)(Hz)

AON=ON(Hz2)

14:025+0:104

1:723+0:014

3:633+0:050

4:306+0:053

1:950+0:016

6:791+0:051

tON=ON(s)

0:235+0:002

0:492+0:006

0:440+0:009

0:327+0:006

0:354+0:004

0:475+0:005

dON=ON(s){0:016+0:001

0:046+0:003{0:022+0:004

0:073+0:003

0:026+0:002

0:008+0:002

2.94

0.63

0.80

2.44

2.06

2.48

SSOFF=OFF T(t)(Hz)

AOFF=OFF

(Hz2)

14:409+0:172

1:808+0:021

3:082+0:050

14:154+0:158

8:641+0:081

21:463+0:138

tOFF=OFF(s)

0:726+0:012

0:729+0:012

0:501+0:011

0:491+0:008

0:798+0:010

0:452+0:004
0:052+0:002

dOFF=OFF(s){0:353+0:006

0:069+0:006{0:362+0:006

0:162+0:004{0:136+0:005

AON=OFF(Hz2)

11:620+0:140

1:561+0:022

6:106+0:091

9:767+0:156

2:638+0:033

10:185+0:084

tON=OFF(s)

0:489+0:008

0:747+0:015

0:428+0:009

0:456+0:010

0:578+0:010

0:556+0:006

dON=OFF(s){1:200+0:004{1:405+0:007{0:765+0:004{1:238+0:005{1:143+0:005{0:890+0:003

The number of ON and OFF cells used in the simulated integrate-and-fire model are also listed for each data set. The OFF/ON correlation function is a reflection about 0of the ON/OFF correlation function, such that AOFF=ON ~AON=OFF , tOFF=ON ~tON=OFF and dOFF=ON ~{dON=OFF .
doi:10.1371/journal.pcbi.1000618.t001

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ON/OFF Segregation in the LGN

Figure 2. STDP versus BTDP and burst detection. (A) STDP modifies synaptic strength based on the timing, Dt~tpost {tpre (indicated byhorizontal arrows), and firing order of pairs of pre- and postsynaptic spikes [10]. Synaptic change occurs in a time window on the order of tens of
milliseconds determined by tz and t{ , with maximum change at Dt&0 given by Az and A{ (%) in EPSC (evoked postsynaptic current) size. (B)BTDP governs synaptic change based on the timing (but not order) of pre- and postsynaptic bursts over a second-long time window, tz [16]. We fit
the blue symmetric exponential curve to the linear experimental fit, where Az and I denote the amounts of maximum synaptic potentiation anddepression in % of EPSC size, respectively. (C) We detected ON (top) and OFF (bottom) bursts by accumulating burst detection variables DON
pre andDOFF
pre . At the arrival of a spike, D increases by 1, and otherwise decays exponentially with a time constant 100 ms. A burst was detected once D
reached a fixed threshold (here 1.5) denoted with an asterisk; D was not allowed to exceed the threshold value. The location of the asterisk (insteadof the start time of the burst) was used to evaluate the burst latency for synaptic change. Parameters for STDP and BTDP are listed in Table 3. Figure
modified from [68] with permission from the HFSP journal.
doi:10.1371/journal.pcbi.1000618.g002

data sets, we computed a segregation index (Equation 16,
Materials and Methods). While a detailed presentation of thesegregation results using each modeling approach and for thewhole range of initial conditions in ½0,wmax  can be made for alldata sets as for data set 1 in Figure 3A, for the sake of brevity we
present only the segregation index with fixed unbiased initial
weights of 4:0 for both ON and OFF in Figure 4A,B. The
segregation index was, in fact, the same for all unbiased initialconditions (uniformly distributed along the main diagonal of thetwo-dimensional weight space) for all data sets as for data set 1 inFigure 3A. The reduced linear model is two-dimensional, thus, bydesign segregation occurs when one weight wins and the otherloses; this always happens if R is sufficiently large. Figure 4A shows
the segregation index under the reduced linear model framework,where the numbers in each bar denote the minimum R required
for the selected unbiased initial conditions to give segregation. A
common requirement for segregation across all the data sets was adepression-to-potentiation ratio R much larger than observed
experimentally [9,10], and in theoretical work with syntheticallygenerated Poisson inputs [24,25]. This was needed to account for
the large pairwise correlations of the real spike recordings, incontrast to the small cross-correlations between independent
Poisson inputs which need R slightly larger than one (for example,
R~1:05) [25].
Unlike the reduced linear model, segregation in simulations
with the integrate-and-fire model is harder to achieve, since it
requires a subset of all weights of one cell type to potentiate
maximally, while all weights of the other cell type to depress. Onlyin data sets 1 (Figure 3A) and 2 (data not shown) we found that
increasing R beyond 3.0 resulted in segregation for a range ofinitial conditions, consistent with the reduced linear model in
Figure 4A. Segregation was not achieved in the other data sets forany studied R. Though it is possible that we did not sample therange of R finely enough, a value of R which could lead toeven a slight dominance of potentiation, thus, suggesting a
depression-to-potentiation ratio less than or equal to 1 (forexample, 0:80 [10], and 0:38 [9]). Under these limitations, westudied STDP with equal potentiation and depression timescales oftz ~t{ ~20 ms, which simplifies the depression-to-potentiation
ratio to the ratio between amplitudes, R~A{ =Az . Fixing the
maximum potentiation amplitude Az , we varied R through the
maximum depression amplitude A{ .
In Figure 3A (theory) we show the weight dynamics for data set
1 for different values of R. The vector fields and example
trajectories demonstrate the segregation outcome for any initialcondition in the range ½0,wmax  for the ON and OFF weights. For
experimentally-observed ratios of R&1:0, any initial condition led
to the maximal weight potentiation of both cell types (top panel).
Increasing R replaced the region where all weights grow, with
regions where depending on initial conditions either ON or OFF
segregation occurred (two middle panels). Many initial condition
combinations (even those giving a prominent bias to ON) resulted
in OFF segregation (bottom panel). We confirmed segregation
results from the reduced linear model by simulating spikes for thepostsynaptic LGN neuron with an integrate-and-fire model in
Figure 3A (simulation), where all ON (3) and OFF (3) spike trainsfrom data set 1 were used as inputs. The space of all initialconditions for the ON and OFF weights in ½0,wmax  was exploredin discrete steps, such that a separate simulation was run for eachcombination of initial conditions. Both theory and simulation
demonstrated a similar trend for increasing R (corresponding
panels in Figure 3A, theory and simulation), though notsurprisingly the ratios did not closely match.
Comparing results from the reduced linear Poisson model andthe simulated integrate-and-fire model for the other data sets (datanot shown), however, did not produce consistent results as for data
set 1. To simultaneously demonstrate the difference in outcomes
between the two models, and the absence of segregation across all
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ON/OFF Segregation in the LGN

Figure 3. STDP without normalization does not result in segregation. (A) STDP with spike integration time windows of 20 ms results in ON/
OFF segregation only for two data sets out of six (data set 1 shown here; data set 2 not shown), assuming a depression-to-potentiation ratio,
R~A{ =Az , much larger than experimentally-observed. Theory and simulation show similar trends, although actual values of R differ. (Theory)
Vector fields and example trajectories of the ON-OFF weight dynamics illustrate results from the linear model. The direction of two eigenvectors of
the plasticity matrix Q (pointing out of the origin for a positive eigenvalue, and into the origin for a negative eigenvalue) determine how R affects
segregation: as R increases, regions of initial conditions where both weights potentiate (trajectories in black) become areas where segregation occurs(trajectories in red for ON and blue for OFF). (Simulation) In the simulated integrate-and-fire model (here, data set 1 with 3 ON and 3 OFF inputs), a
separate simulation was run for each initial condition uniformly sampled between 0 and the maximum weight value, wmax , for the ON and the OFF
weights. Here wmax ~5:0 and the discrete steps for the initial weights were 0:5. The colored symbol indicates the segregation outcome according to
the legend, matching the color of the trajectories in the linear model. The size of the colored circles denotes the percentage of synaptic weights of aparticular RGC type which potentiated maximally out of all RGCs of the same type initially wired to the LGN neuron (the three dots denote 33, 66 and100%, respectively), while all weights of the other RGC type depressed to 0. Initial weights too small to generate postsynaptic activity result in no
synaptic change (dots in the bottom left region of each plot). (B) Extending the spike integration window to match the timescale of the input
correlations to 500 ms, results in pure dominance of the ON cells, both under the theoretical and the simulated model and for any value of the ratio R(data set 1 shown here, but all others show the same behavior).
doi:10.1371/journal.pcbi.1000618.g003

the STDP integral, A{ t{ =Az tz , took larger values by considering
smaller ratios of the depression and potentiation amplitudes,
R~A{ =Az . However, only data set 1 resulted in OFF segregation
for this choice of STDP parameters (data not shown).
In conclusion, our modeling showed that a temporallyasymmetric plasticity rule like STDP, integrating pre- andpostsynaptic spikes over short timescales on the order of tens of
milliseconds which ignore the correlation timescales between
inputs of different cell type, failed to segregate these ON and OFF
RGC inputs in a model of a developing LGN over a wide range ofparameters and in the absence of synaptic competition. Under
STDP, the growth of synaptic weights of one cell type did notprevent the growth of weights of the other cell type, because ON
and OFF cells fire independently within the timescale of STDP.

segregation would require careful tuning because of two
observations: (i) increasing R beyond some value eventually led
to depression of all weights, and (ii) for smaller values of R for
which we did observe potentiation of some weights, the result wasalways potentiation of both ON and OFF weights as these tightly
correlated RGCs of different cell type reliably drove postsynaptic
spikes within 5–10 ms. A summary of the segregation index
simultaneously showing the absence of segregation for all data sets
is given in Figure 4B for a sample ratio of R~2:0 and unbiased
initial conditions of 4:0 for both ON and OFF weights.
We also explored an STDP rule with a longer temporal window
for synaptic depression than for potentiation corresponding toexperiments from somatosensory cortex, tz ~10 ms andt{ ~60 ms [26]. Now the ratio of the negative to positive areas of
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Figure 4. A summary of segregation under STDP and BTDP. (A) Segregation indices using Equation 16 for all data sets following the linear
modeling approach with STDP using unbiased initial conditions of 4.0 for both ON and OFF weights. By design the theoretical model always results insegregation for large enough R; the minimum R for segregation is written in each bar. (B) Segregation indices from numerically implementing STDP
for all data sets illustrate the absence of segregation. A high R~2:0 was used here; increasing R beyond 3.0 results in segregation only for data sets 1(Figure 2A) and 2 (data not shown), inconsistent with linear model predictions in (A). (C) Segregation indices from the theoretical and numerical
implementation of BTDP show consistent segregation across all data sets using the experimentally-observed ratio R~I=Az ~0:42 [16]. (D) STDP
with subtractive normalization can induce weight competition and segregation in the linear model. For realistic depression-to-potentiation values ofR, the outcome is always OFF segregation. Increasing R to the values which generated segregation in (A) (denoted by the black vertical lines) resultsin ON segregation for sets 4 and 6, matching results from (A). Note that in the linear model wmax simply scales the weights and thus, it does not affectsegregation outcome. (E) STDP with subtractive normalization also results in segregation in the simulated integrate-and-fire model for realistic valuesof R. However, results depend on parameters: for instance, changes in R (horizontal axis) sometimes result in a switch from ON to OFF segregation(data set 2), and sometimes from ON to no segregation (data set 3). Also, the amount of total synaptic weight maintained by the postsynaptic neuron(top bar for each set wmax ~5 and bottom bar wmax ~10) affects segregation outcome (for example, data sets 4 and 5). While subtractive
normalization rescues segregation for the STDP rule, results are inconsistent across the two different modeling approaches (compare to D).
Uncolored sections in each bar denote no segregation.
doi:10.1371/journal.pcbi.1000618.g004

results are not too surprising given recent theoretical work on
STDP [31]. Introducing dependence of weight change on thecurrent weight, choosing a different spike-pairing scheme or
implementing dendritic and axonal synaptic delays may differently
affect the behavior of the modeled system [31]. Furthermore,existing correlation-based models of ocular dominance require
some form of synaptic competition to segregate inputs from the left
and the right eye [32], which we considered next.

Therefore, tightly correlated groups of ON and OFF cells which
effectively drive the postsynaptic neuron, required physiologicallyunrealistic, carefully-tuned values of the depression-to-potentiation
ratio, R, for segregation. In the integrate-and-fire model even large
R could not rescue segregation, because multiple ON and OFF
inputs caused reliable spiking of the LGN neuron, and large R
resulted in nonselective competition between all ON and OFF
cells, rather than between groups of ON and OFF cells. These
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ON/OFF Segregation in the LGN

correlations with STDP in Equation 8 (Materials and Methods) is
significantly negative for large R for all cell pairs, and moreover, it
is more negative for same cell types due to their larger correlations
in comparison to opposite cell types. Therefore, for some data sets
where the negative contribution of the correlations dominated thepositive contribution of the average firing rates (data sets 4 and 6),
the ON cell with lower firing rate won for sufficiently large R. In
particular, for R equal to the values which resulted in segregation
without normalization in Figure 4A (indicated by the short verticallines in Figure 4D), data sets 4 and 6 produced ON segregationconsistent with Figure 4A.
In the nonlinear integrate-and-fire model where multiple ON
and OFF inputs were used, implementation of homeostasis also
rescued segregation for biologically-plausible values of R (Figure 4E).
However, even with subtractive normalization the final outcome insimulations depended on parameter choice (depression-to-potentiation ratio R and maximum synaptic strength wmax ) and differedfrom the outcome of the reduced linear model (Figure 4D,E andTable 2). To understand whether the dependence of the segregation
outcome was the result of variable input statistics, or the
nonlinearity of the integrate-and-fire model, we simulated the full
linear model using all RGCs in each data set. In Table 2 wesummarize the segregation outcome for a realistic depression-topotentiation ratio R between 0.8 and 1.0, under the three different
models: the reduced linear model (Figure 4D), the full linear model,
and the integrate-and-fire model, for a range of different wmax
values (Figure 4E). Both the reduced and the full linear model had apreference for OFF segregation for all data sets. Above, we
explained that the reason for this OFF bias is due to the diminishingeffect of the correlations under STDP and the dominance of firing
rates (which are generally higher for the OFF cells) for R&1. As an
exception, the full linear model for data set 3 failed to capture OFF
segregation because the multiple ON and OFF cells had morevariable firing rates than the consistently larger OFF firing rates in
the other data sets. With the output nonlinearity of the integrateand-fire model, we observed two regimes: (1) for a small wmax , theON cells won in all data sets (Table 2) unlike the linear model,
probably because the short timescales of STDP captured the highinstantaneous firing rates of ON cells within a burst (as indicated by
the smaller decay timescales for ON/ON than OFF/OFF RGC
pair correlations in Table 1). The OFF cells most likely failed todrive the postsynaptic neuron as efficiently as the ON cells, due theirlower instantaneous firing rates within a burst; (2) for a large wmax ,the OFF cells won in most data sets in agreement with the linear
model. The exact transition point from (1) to (2) depended on thedata set (Table 2). Thus, fine-tuning of wmax is necessary to develop
similar numbers of ON-selective and OFF-selective LGN neurons.
Again, the sensitivity to wmax arises because STDP does not exploit
the correlations between cells, but rather the time-averaged firing
rates. In the regime of R&1, the contributions of the potentiationand depression regions of STDP cancel with each other yielding
dominance of the time-averaged firing rates and OFF segregation.
In the less-relevant regime of Rw2, STDP suppresses cooperation
between same type cells. While in the reduced linear model thisincrease in R resulted in a switch from OFF to ON segregation fordata sets 4 and 6, in the full linear model and in the simulated
integrate-and-fire model, it introduced stronger competition
between same type than between opposite type RGCs, eventuallyeliminating segregation (Figure 4E, large R).
In summary, we showed that adding subtractive normalization
to STDP resulted in ON/OFF segregation but the outcome was
highly sensitive to the choice of parameters. For biologicallyrealistic depression-to-potentiation values of R&1, temporallyasymmetric STDP with normalization consistently favored one cell

STDP with homeostatic mechanisms can rescue
segregation in carefully-tuned regimes
It is likely that synaptic plasticity rules work together withhomeostatic mechanisms during formation and refinement ofdeveloping circuits [33]. While most work has examineddeveloping cortical neurons where synaptic scaling is induced in
response to changes in global activity levels [34,35], there is recent
evidence of response homeostasis at the developing retinocollicularsystem maintained at the level of synapses [36]. We considered theimplications of this second type of homeostasis together with
STDP on segregation. Chandrasekaran et al. [36] found that at
the level of a single postsynaptic neuron in the superior colliculus,
the total number and strength of individual synapses is preserved
during development. We implemented this as subtractive normalization, because it has been shown to induce weight competition(in contrast to divisive normalization) in traditional forms of
Hebbian learning [27,37]. For each data set, we chose a numberof synaptic weights N that the postsynaptic LGN neuron would
maintain at adulthood with maximal strength wmax , such that the
total synaptic input to the LGN neuron maintained at all time
during the simulation was Nwmax . In addition to STDP, synaptic
weights were modified by adding/subtracting the synaptic deficit/
excess towards maintaining the target synaptic strength (Materials
and Methods). As in STDP with large R (Figure 4A), STDP withsubtractive normalization for the reduced linear model where asingle ON and a single OFF weight competed, also resulted insegregation but for biologically-plausible values of R (Figure 4D).
However, the segregation outcome for R&1:0 in Figure 4D did
not match results from STDP without normalization in Figure 4A,
and in particular the outcome was always OFF segregation(Table 2). While same cell type correlations are higher than
opposite cell type correlations, when considered within the
millisecond-long window of STDP these second-long correlations
are noisy and essentially constant. Then for biologically-plausible
ratios of depression-to-potentation near unity (R&1) the effect of
the correlations is diminished, because the contributions from thepotentiating and from the depressing part of temporally-asymmetric STDP cancel. In this case, the evolution of the weights is
entirely determined by single neuron properties, and in particular,
by the time-averaged firing rate of each cell. The OFF cells for alldata sets have larger firing rates than the ON cells (Table 1),leading to a bias for OFF segregation in Figure 4D. As R increases,
the correlations start to become more relevant. The product of theTable 2. Parameter sensitivity of STDP with subtractive
normalization.

Model

set 1

set 2

set 3

set 4

set 5

set 6

2D linear

OFF

OFF

OFF

OFF

OFF

OFF

full linear

OFF

OFF

—

OFF

OFF

OFF

I&F with wmax ~5

ON

ON

ON

ON

ON

ON

I&F with wmax ~8

OFF

ON

ON

ON

OFF

ON

I&F with wmax ~10

OFF

ON

ON

OFF

OFF

—

I&F with wmax ~15

OFF

OFF

ON

OFF

OFF

—

I&F with wmax ~20

—

OFF

—

OFF

OFF

—

Segregation results are shown for the reduced two-dimensional linear model,
the full linear model with all RGC inputs, and the nonlinear integrate-and-firemodel with all RGC inputs for a biologically-plausible range of the depressionto-potentiation ratio R[½0:8,1. ON denotes ON segregation, OFF denotes OFF
segregation, and a dash denotes no segregation.
doi:10.1371/journal.pcbi.1000618.t002

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ON/OFF Segregation in the LGN

activity over the second-long correlation timescales, must do so
without giving a naı̈ve advantage to the cell which fires first.
STDP, on the contrary, favors synaptic inputs that can serve as‘earliest predictors’ of other spike events [25,40]. Now we turn to
BTDP which integrates pre- and postsynaptic bursts irrespective of
their firing order.

type over the other, depending on the model of the postsynaptic
neuron. We determined a parameter range for wmax where thelinear model can approximate the nonlinear model, but to obtain
an equal preference for ON or OFF segregation, fine-tuning of
wmax was critical. Even though it is plausible that the sensitivity ofthe model to various parameters may be implemented by abiological system during development, the model predicts that thesystem will produce very different outcomes for any small
perturbation induced by environmental changes, stochastic events,
unreliable vesicle release, and so on, which seems unlikely. The
millisecond-long integration timescales of STDP are too short toreliably sample the noisy correlation functions which naturally
extend over much longer timescales, thus making STDP
unsuitable to study segregation in this system. Later in the paper
we explore a more appropriate plasticity rule modulating synaptic
change and during development, which resulted in segregation
without the need for presynaptic homeostatic control or carefulparameter tuning (results summarized in Figure 4C).

Temporally-symmetric BTDP with second-long timescales
robustly guides ON/OFF segregation
Can a temporally-symmetric burst-based rule integrating preand postsynaptic bursts over second-long timescales guide ON/
OFF segregation in the LGN? Such a rule, BTDP, wasexperimentally-proposed for the developing retinogeniculate
system and tested in a model for eye-specific segregation using
simulated retinal waves [16]. We found that BTDP robustly guides
ON/OFF segregation in all data sets (Figure 5), without requiring
homeostatic control of presynaptic connectivity to introduce
synaptic competition. BTDP resulted in segregation assuming arealistic, experimentally-observed ratio of depression-to-potentiation R~I=Az (Figure 2B). (Although R is defined differently for
STDP and BTDP, it has the same meaning in both rules.)Segregation outcome fell under two qualitative trends: dominanceof ON segregation (data sets 1–3 in Dataset S1, Figure 5A) anddominance of OFF segregation (data sets 4–6 in Dataset S1,Figure 5B), where ‘dominance’ was defined as which cell typenormally won, averaged over all initial conditions.
In Figure 5A (theory) we show the segregation outcome under
the reduced linear model which used as inputs the correlation
functions (Table 1 and Figure 1B) for data set 1, as a representativeof data sets 1–3. Fixing the symmetric timescale of integration tz
and the maximum amplitude of potentiation Az according toTable 3, we varied the depression-to-potentiation ratio R~I=Az
by changing the amount of depression I. As R increased towards
the experimentally-observed value of 0.42 [16], initial conditions
which originally led to the maximal potentiation of both weights(top panel), now resulted in ON or OFF segregation (two middle
panels). ON segregation was the only outcome for large R (bottom
panel). Further increasing R led to overall depression of all weights(data not shown). The results with the reduced linear model driven
by the correlation functions of the most correlated ON and OFF
cell pairs were confirmed when simulating spiking activity using anonlinear integrate-and-fire neuron for the LGN neuron driven by
all ON (3) and all OFF (3) spikes trains in data set 1 (Figure 5A,
simulation). For each combination of initial conditions of ON and
OFF weights and experimentally-observed ratio R&0:42 [16],some weights of only one cell type potentiated, while all weights of
the other type depressed, indicating successful segregation(Figure 5A, simulation, third panel). Since there were different
numbers of ON and OFF inputs in each data set (Table 1), ourresults show that segregation is robust and is not biased by the
initial numbers of ON and OFF RGCs connecting to the LGN
neuron. Furthermore, the match between any R of the reduced
linear and the simulated integrate-and-fire model was remarkably
good. We called the trend illustrated in Figure 5A dominance of
ON segregation, because for the experimentally-observed ratio R[16], more initial condition combinations resulted in ON
segregation than in OFF segregation (Figure 5A, third panels).
In particular, unbiased initial conditions for the ON and OFF
weights (located along the main diagonal), and even a small OFF
bias in initial conditions, resulted in ON segregation. Data sets 2
and 3 showed qualitatively similar dominance of ON segregation(data not shown).
The segregation outcome under the reduced linear model fordata set 4, as a representative of data sets 4–6, is illustrated in

Temporally-asymmetric STDP with second-longtimescales favors ON segregation
A spike-based rule like STDP cannot explain ON/OFF
segregation in the LGN without synaptic competition because
tightly correlated retinal inputs of different cell type reliably drive
postsynaptic activity within 5–10 ms. Even with subtractive
normalization, the short timescales of STDP are ineffective atrobustly driving ON/OFF segregation. Butts and Rokhsar [14]found that most information content of spontaneous retinal waves
is contained over timescales of 100–1000 ms. Furthermore, Butts
et al. [16] recently proposed a burst-based rule for modifying
synaptic strength over second-long timescales in the developing
retinogeniculate system. We next asked if any plasticity rule
integrating pre- and postsynaptic activity over the relevant input
correlation timescales of ON and OFF RGCs can explain
segregation. Thus, we investigated a second plasticity rule, amodified STDP with extended spike integration timescales.
Vector fields and example trajectories in Figure 3B (theory)
illustrate segregation scenarios produced with the reduced linear
model for data set 1 with tz ~t{ ~500 ms in STDP. We
confirmed these in Figure 3B (simulation) with the integrate-andfire model where all ON (3) and OFF (3) inputs in data set 1 wereused with multiple (discretely-sampled) initial conditions in½0,wmax  for the ON and OFF weights. As the depression-topotentiation ratio R increased, the region of initial conditions forwhich weights of both cell types potentiated was graduallyreplaced with a region of ON segregation. Segregation results
followed a similar qualitative trend for the other five data sets (datanot shown). In conclusion, modifying synaptic weights based on
STDP with second-long timescales for spike integration to match
input correlation timescales resulted in ON segregation for all data
sets and for any combination of initial conditions. This effect was
not only produced by the reduced linear model, but also by thesimulated integrate-and-fire model for the LGN neuron. Despite
different numbers of ON and OFF retinal inputs in each data set(Table 1), a subset of ON synaptic weights always potentiated,while all OFF weights were eliminated. Since LGN neurons can beboth either ON- or OFF-responsive [17,38,39], we conclude thatsuch a rule is implausible as it never results in OFF-responsive
LGN neurons.
These modeling studies demonstrate that a plasticity rule forsegregation does not only have to integrate pre- and postsynaptic
activity over timescales matching those of the input correlations,but that other constraints are also required. Since the firing of ON
RGCs precedes that of OFF RGCs, any rule which integrates
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ON/OFF Segregation in the LGN

Figure 5. BTDP results in robust segregation. (A) Segregation results using BTDP for data set 1, a representative spike train recording of data
sets 1–3 which have similar peaks of the pairwise input correlation for ON/ON and OFF/OFF pairs. Drawing conventions and legend as in Figure 2, andas before wmax ~5:0 with discrete steps for the initial weights of 0:5. As the depression-to-potentiation ratio in BTDP, R~I=Az , increases towards
the experimentally-observed ratio, R&0:42 [16], theory and simulation show the segregation of ON and OFF RGC inputs (red and blue trajectories),
emerging from a state where the weights of both cell types potentiate maximally for small R (black trajectories). The region of initial conditions
resulting in ON segregation is larger than the region resulting in OFF segregation, indicative of ON dominance (see main text). Similarly, unbiased
initial conditions located along the main diagonal in each plot show ON segregation. (B) Segregation results using BTDP for data set 4, a
representative spike train recording of data sets 4–6, which have higher OFF/OFF input correlation peaks than ON/ON peaks. As R increases, thedominant segregation outcome for a larger set of initial conditions is OFF segregation, in contrast to (A). While the match between theory andsimulation is consistent for data sets 4–6 and experimentally-observed R [16], for data sets 4 and 5 a narrow range of large ratios (denoted by R )resulted in OFF segregation only in simulations (bottom panels). To match predictions of the two models for this larger than experimentally-observed
R, tOFF was made larger than tON in the theoretical model (from 0.327 to 0.500, Table 1).
doi:10.1371/journal.pcbi.1000618.g005

Figure 5B (theory). As before, we increased R towards theexperimentally-observed value of 0:42 [16] and saw that initialconditions which led to the maximal potentiation of both weights
for small R (top panel), now resulted in ON or OFF segregation(two middle panels). To obtain OFF segregation as the only
outcome for R larger than experimentally-observed in the same
way we obtained ON segregation for data sets 1–3 in Figure 5A(bottom panels), we had to extend the correlation timescale(tON=ON ) of ON RGC pairs in Figure 5B (theory, bottom panel).
Simulating spiking activity using a nonlinear integrate-and-firemodel for the LGN neuron driven by all ON (5) and all OFF (2)spike trains in data set 4 matched the segregation results from thereduced linear model, with a close correspondence of R (Figure 5B,
simulation). Later we discuss why for large R the simulated model,
but not the theoretical model, resulted in OFF segregation for all
initial conditions. As before, segregation was robust and did notdepend on the initial numbers of ON and OFF retinal inputs
PLoS Computational Biology | www.ploscompbiol.org(Table 1). Data sets 4–6 showed dominance of OFF segregation,
for the experimentally-observed ratio R [16] shown in Figure 5B(third panels). Furthermore, unbiased initial conditions for the ON
and OFF weights (located along the main diagonal), and even asmall ON bias in initial conditions, resulted in OFF segregation.
Data sets 5 and 6 showed qualitatively similar dominance of OFF
segregation (data not shown).
To summarize, we found that not only the second-longtimescales of bursts integration are needed for segregation, butalso the temporally-symmetric feature of BTDP. In particular, thenegative and symmetric BTDP window at long temporal delays infiring, mutually inhibited the simultaneous growth of ON and
OFF synaptic weights, thus providing the necessary synaptic
competition without implementing additional homeostatic mechanisms, in contrast to STDP. In Figure 4C, we summarize thesegregation results of BTDP for all the data sets using experimental
values of R [16], and unbiased initial conditions. Figures 4C and9

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ON/OFF Segregation in the LGN

simulations with BTDP using wmax ~5 (Figure 6A) and wmax ~20(Figure 6B). We observed no differences in the segregation
outcome for initial conditions in the region ½0,5 when both ON
and OFF segregation occur, in contrast to STDP with subtractive
normalization (Figure 4E). The main effect of increasing wmax wasthat simulations took longer to show segregation. Therefore,
segregation results with BTDP are robust under changes in thescaling of synaptic weights. Note that the linear model is by design
independent of wmax , which supports its usefulness in predicting
segregation.

Table 3. Model parameters for generation of LGN activity
using the linear model (theory) and the integrate-and-firemodel (simulation), and for synaptic modification induced by
STDP or BTDP.

Parameter

NotationValue

t1 (ms)

10

TheoryEPSP kernel

100
t2 (ms)The one-second offset in the firing of ON and OFF RGCs
is necessary for segregation

550

STDP/BTDP

Az

0.001

a

0.02

b

0.2

Kerschensteiner and Wong [20] hypothesized that ON and
OFF RGCs segregate in the LGN because of the one-second
temporal offset in their precisely-ordered firing. We performed
simulations with BTDP with the integrate-and-fire model in which
we shifted the firing of the ON cells by one second such that it wassynchronous with the OFF cells, and found that this abolishedsegregation (Figure 6C). Analyzing the reduced linear model
where the ON/OFF and OFF/ON correlation functions were
shifted such that the peak occurred at *0 ms, also eliminatedsegregation (data not shown). This confirms that the temporal
asynchrony is a necessary activity cue for the segregation of RGCs
of different type, but not a sufficient cue. As we demonstrate next,
the correlation structure of the inputs also significantly affects thesegregation outcome.

Simulation
Integrate-and-fire

STDP/BTDP

c

250

d

2

Vth

30

Az

0.00050.0001

Common parameters
Upper synaptic bound

wmax

5
10

STDP



tz (ms)20

Insights from the linear model: eigenvalue analysis

20

We showed that BTDP can explain ON/OFF segregation in
the developing LGN of mouse by integrating activity (i) overtimescales relevant to the inputs, and (ii) irrespective of the order of
pre- and postsynaptic activity. Furthermore, half of the studied
data sets (1–3) demonstrated dominance of ON segregation, andthe other half (sets 4–6) demonstrated dominance of OFF
segregation. To understand why BTDP successfully captured
segregation without additional homeostatic mechanisms (but notstandard STDP, nor STDP with extended timescales), and to
determine which features of the inputs specified ON versus OFF
dominance, we dissected the linear model of Equation 7 (Materials
and Methods).
For comparison, in STDP with subtractive normalization the
RGC firing rates dominated segregation due to the millisecondlong integration timescales, while the contribution from thecorrelations cancelled due to the temporal asymmetry of STDP.
In BTDP, however, we show that the RGC correlations dominate
segregation due to the matching second-long integration timescales, and are further intensified by the temporal symmetry of
BTDP. The entries in the plasticity matrix Q of Equation 8(Materials and Methods) can be obtained by multiplying the area
under the input correlation functions (Figure 1B) and the area
under BTDP (Figure 2B). From the decaying exponential fits of
the input correlations (Figure 1B and Table 1), we extracted acommon feature among the six data sets to be a smaller decay
timescale for ON/ON pairs than for OFF/OFF pairs (for data set
6 they are approximately equal), suggesting that ON cells fire
shorter high-frequency bursts. On the other hand, after comparing
correlation peaks for pairs of different cell types, we found that thesix data sets form two groups, which coincide with the preference
for ON or OFF dominance (Figure 4C). Data sets 1–3 have similarcorrelation peaks for ON/ON and OFF/OFF cell pairs(Figure 7A, left), while data sets 4–6 have a higher correlation
peak for OFF/OFF pairs than for ON/ON pairs (Figure 7A,
middle). Correlation functions for different type RGC pairs are

10500t{ (ms)

20
60
500

BTDP

tz (ms)500

Multiple values were tested for some parameters, marked by asterisks.
doi:10.1371/journal.pcbi.1000618.t003

5A,B demonstrate that the structure of the inputs, in combination
with initial conditions for the ON and OFF weights, determine thesegregation outcome such that data sets 1–3 showed dominance of
ON segregation, while data sets 4–6 dominance of OFF
segregation. Mathematical interpretation of the result based on
this simple model is explored later in this paper.

Upper bound for synaptic weights does not affectsegregation under BTDP
The ratio of final (wmax ) to initial synaptic strength of thewinning cell type after segregation indicates the total amount ofsynaptic strengthening during this stage of development. Chen andRegehr [6] showed that in addition to the pruning of RGC inputs
converging to a single LGN neuron (from w20 inputs at birth, to
1–3 after eye-opening), synaptic weights strengthen *50-fold.
According to more recent estimates, 5–8 input RGCs before theonset of glutamatergic waves refine to a few after eye-opening, butthese studies did not report the total amount of synaptic
strengthening during glutamatergic waves [41,42]. So far we
explored the segregation outcome with BTDP for initial conditions
uniformly distributed in the range ½0,wmax  for wmax ~5. To study
the sensitivity to the upper weight bound wmax , we compared
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ON/OFF Segregation in the LGN

Figure 6. Features which guide segregation under BTDP. (A) Temporal evolution of synaptic weights with R~I=Az ~0:30 for data set 1 (3ON and 3 OFF cells), as in Figure 4A, for initial conditions when the ON (first column) or the OFF cells (second column) segregate. Similarly, sample
trajectories for data set 4 (5 ON and 2 OFF cells), as in Figure 4B, for initial conditions where the ON (third column) or the OFF cells (fourth column)
segregate. Upper bound on the weights wmax ~5. Even though there are more ON than OFF inputs in data set 4, the outcome does not depend on
the number of ON and OFF inputs, but on the initial conditions of the synaptic weights. Furthermore, a simple bias in the initial conditions does notalways result in segregation towards the biased cell (see Figure 4 for the segregation outcome for all initial condition combinations). (B) Increasing
the upper bound of the weights to wmax ~20 does not affect the segregation outcome; weights simply take longer to segregate. (C) Delaying thetime of the spikes of ON cells by 1 second, such that they are synchronous with OFF cells, eliminates segregation.
doi:10.1371/journal.pcbi.1000618.g006

the other eigenvector, ½z,zT , corresponding to l2 andpotentiation of both weights, is suppressed (here, z and {indicate positive and negative entries in each eigenvector, and T
denotes transpose). Segregation outcome depends on the relative
size of the main diagonal terms in Q. For data sets 1–3 where thepeaks for ON/ON and OFF/OFF cell pairs are similar, a larger
area of the OFF/OFF correlation (than of the ON/ON
correlation) falls under the negative part of BTDP (Figure 7A,left). This gives qOFF=OFF vqON=ON and resulted in dominance of
ON segregation in our simulations (Figure 4C). For data sets 4–6,
with larger OFF/OFF correlation peaks compared to ON/ON
correlation peaks, a larger area of the OFF/OFF correlation (thanof the ON/ON correlation) falls under the positive part of BTDP
(Figure 7A, middle). This gives qOFF=OFF wqON=ON and resulted indominance of OFF segregation in our simulations (Figure 4C).
Even though this interpretation of the reduced linear model
does not directly use the recorded spike trains in each data set(whereas the simulated integrate-and-fire model does), but instead
uses fitted estimates of the correlations computed from the mostcorrelated cell pairs, results from the linear and integrate-and-firemodels agree (Figure 5). We observed an inconsistency only indata sets 4 and 5 for R larger than experimentally-observed, where
the original correlation peaks and timescales from Table 1 resulted
in dominance of ON segregation (data not shown). In this case, thereflections of each other about 0 seconds and have a peak locatedat +1 second (Figure 7A, right). Therefore, a significantly larger
area of the correlation function of different type RGC pairs falls
under the negative part (than under the positive part) of BTDP
(Figure 7A, right). This results in negative off-diagonal entries in
the plasticity matrix Q of Equation 8 (Materials and Methods),
which are equal due to the reflection about 0 seconds,
qON=OFF ~qOFF=ON ~xv0:"
Q~

qON=ONxx

qOFF=OFF#:ð1Þ

A matrix of this form has eigenvalues1
qON=ON zqOFF=OFF2
ﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
1 
qON=ON {qOFF=OFF {4x2 [ R+2

l1,2 ~

ð2Þ

where l1 wl2 (l1 corresponds to the expression with ‘z’ and l2
with ‘{’ sign). The eigenvector, ½z,{T , corresponding to thelarger eigenvalue l1 and segregation of the weights, is dominant;
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ON/OFF Segregation in the LGN

Figure 7. Insights into ON/OFF segregation from the linear model. (A) (left) ON dominance. Factors which determine segregation for data
sets 1–3 (data set 1 in Figure 4A) with similar correlation peaks of ON/ON and OFF/OFF pairs. Segregation is the result of a trade-off between therelative areas of the input correlation functions for ON/ON (red) and OFF/OFF (dark blue) pairs integrated by the positive and negative parts of BTDP
(green). BTDP favors ON segregation because the smaller area of the ON/ON correlation under the negative part of BTDP (denoted by {) dominates
the larger area of the OFF/OFF correlation under the positive part of BTDP (denoted by z). Therefore, qON=ON wqOFF=OFF in the plasticity matrix Q(Equation 1). (middle) OFF dominance. Factors which determine segregation for data sets 4–6 (data set 4 in Figure 4B) with larger correlation peaks of
OFF/OFF than ON/ON pairs. BTDP favors OFF segregation because the larger area of the OFF/OFF correlation under the positive part of BTDP
dominates the smaller area of the ON-ON correlation under the negative part of BTDP, such that qON=ON vqOFF=OFF in the plasticity matrix Q(Equation 1). (right) The reflected (about 0 seconds) input correlations between cells of different type (light blue) have a larger part of their areas
under the negative part of BTDP, ensuring negative off-diagonal terms x in Q (Equation 1) and introducing competition. (B) (left) Input correlation
functions for two of the 15 ferret data sets illustrate dominance of ON segregation. (middle) Input correlation functions for the remaining 13 ferretdata sets illustrate dominance of OFF segregation. (right) The input correlation function between cells of different type peaks at 0 seconds for ferret,
however, the wide correlation timescale (wider than in mouse) still produces negative off-diagonal terms x in Q (Equation 1), and correspondingly,
competition between the ON and OFF weights. (C) STDP drawn to scale to compare its spike integration time window with the timescale of input
correlations shown in (A) and (B). The near-0 off-diagonal terms in Q (Equation 3) inhibit segregation.
doi:10.1371/journal.pcbi.1000618.g007

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ON/OFF Segregation in the LGN

for the wiring of the LGN neuron, are located close to each other(within 50 mm). Thus, it is possible that RGC axons which lose thecompetition at one LGN neuron because of stronger competitors,
win at another. This developmental mechanism is similar to thatseen in mouse neuromuscular junction where the fate of axonal
branches is dependent on the identity of the axons with which they
compete [43].

larger area of the OFF/OFF correlation (than of the ON/ON
correlation) falling under the positive part of BTDP, is not large
enough to counteract the larger area of the OFF/OFF correlation(than of the ON/ON correlation) falling under the negative part of
BTDP. To achieve the expected dominance of OFF segregation,
we had to increase the decay timescale of the ON/ON correlation(Figure 5B, theory, bottom panel). The simulated integrate-andfire model using the recorded spike trains (instead of the ON/ON
correlations) successfully captured segregation with OFF dominance (Figure 5B, simulation, bottom panel). This suggests that
real spike trains may contain higher-order moments which are notalways captured by the pairwise correlations of Figure 1B, or thatsome nonlinear effects of the integrate-and-fire model dominate
segregation outcome. Note that this mismatch between the
reduced linear and the integrate-and-fire model occurred only
for a small range of R larger than experimentally-observed, andfor only two of the six data sets.
In Figure 7C we show a sketch of standard STDP to illustrate
the contrast in timescales of spike integration and input
correlations. STDP ignores the peaks of the ON/OFF and
OFF/ON correlation functions located at +1 second, and theresulting plasticity matrix has off-diagonal entries close to 0 (for
R&1)"

qON=ONQ~
00
qOFF=OFF

ON/OFF segregation in ferret
The BTDP model we just described suggests how spontaneous
retinal waves can instruct ON/OFF segregation in individual
LGN neurons in mouse. Is this result specific to mouse RGCs, ormight this model explain segregation in other systems? The firingproperties during retinal waves in ferret significantly differ fromthose in mouse [22]. During the period of ON/OFF segregation inferret, RGCs of different type fire synchronously, with OFF cells
having a significantly higher firing rate than ON cells [21,22]. In
addition, we found that pairs of different type in ferret are
correlated over longer timescales than pairs of different type inmouse. In a Hebbian covariance-based model with presynaptic
threshold, Lee et al. [22] showed how these patterns can instruct
ON/OFF segregation in ferret LGN. We demonstrate that ourmodel with BTDP can also explain ON/OFF segregation in ferret
and that it is consistent with the model of Lee et al. [22] (Text S1).
Using the recorded spike trains from the Lee at al. study (15 setsin Dataset S1, each consisting of one ON and one OFF RGC spiketrain recorded with single electrodes) [22], we simulated the spikesof a postsynaptic LGN neuron using an integrate-and-fire model.
As before, synaptic modification was subject to STDP or BTDP,
and synaptic weights were bounded between 0 and wmax . In Table
S1, we list the segregation outcome using each plasticity rule.
Segregation was more easily achieved than in the mouse model as
we only used one ON and one OFF RGC. Thus, as in the reduced
linear model, we saw that standard STDP required large ratios ofdepression-to-potentiation to capture segregation. In contrast,
BTDP successfully interpreted firing patterns into segregation forexperimental parameters ranges [16].
Earlier we showed that BTDP failed to induce ON/OFF
segregation in mice when the one-second firing offset between the
ON and OFF cells was eliminated. Although ON and OFF cells firesynchronously in ferret, BTDP could still induce segregation
because the correlation between cells of different type in ferret falls
off much slower than the correlation between cells of different typein mouse (Figure 7A,B, right and Table 1; also Table S2). We
applied a similar analysis to the linear model in Equation 7(Materials and Methods) as in the previous section. The plasticity
matrix Q is similar to that for mice with BTDP (Equation 1), withequal and negative off-diagonal entries because the long tails of the
ON/OFF correlation function of ferret fall under the negative partof BTDP. As before, the relative sizes of the main diagonal terms in
the plasticity matrix Q, qON=ON and qOFF=OFF , determine thedominance of ON or OFF segregation. Since we only have one ON
and one OFF RGCs in each data set, qON=ON and qOFF=OFF can becomputed from the ON and OFF autocorrelations. In two of the 15
data sets (the outliers) we saw a higher correlation peak of the ON
autocorrelation (Figure 7B, left), and in the rest of the sets a higher
correlation peak of the OFF autocorrelation function (Figure 7B,
middle), indicative of the higher firing rate of OFF RGCs (Table
S2). In Text S1 we show how our model with BTDP corresponds tothe covariance-based model used in Lee et al. [22]. In particular, we
make a correspondence between R (the depression-to-potentiation
ratio in our model with BTDP), and h (the presynaptic threshold in[22]) which induced competition between the ON and OFF
weights. Due to the higher autocorrelations of OFF RGCs, our

#
:

ð3Þ

With eigenvectors equal to ½z,0T and ½0,zT , the reduced linear
model offers a simple explanation about why STDP failed tocapture segregation for R&1 and without normalization constraints. Note that the main diagonal entries are large and positive,
due to the contribution from the autocorrelations in Equation 8 for
Q (Materials and Methods), as discussed previously. Increasing R
introduced the required asymmetry for competition between thetwo weights. This follows from the current analysis because large
R makes the off-diagonal entries in Q negative as in Equation 1.
Extending integration timescales of STDP to the relevant input
correlation timescales results in the plasticity matrix"
Q~

qON=ONz{

qOFF=OFF#,ð4Þ

where qON=OFF w0 since the ON/OFF correlation peaks at
z1 second, while qOFF=ON v0 since the OFF/ON correlation
peaks at {1 second. This analysis of the modified STDP rule
explains why it always resulted in ON segregation in our
simulations.
In summary, we have shown that BTDP can explain ON/OFF
segregation in the developing mouse LGN driven by spontaneous
retinal waves. BTDP promoted cooperation between weights ofsame type cells and competition between weights of different typecells without additional homeostatic control. We predicted that the
LGN neuron will become ON- or OFF-responsive based on a
local competition of the firing patterns of neighboring RGCs
connecting to it, whose most relevant features for segregation arecontained in the pairwise temporal correlations. As ON bursts areshorter than OFF bursts as suggested by the smaller correlation
timescales, under BTDP this would suggest dominance of ON
segregation. However, if OFF bursts are more highly correlated assuggested by the higher correlation peaks, then the result isdominance of OFF segregation. In particular, we saw that ON and
OFF RGCs with sufficiently-different firing properties to compete
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ON/OFF Segregation in the LGN

model similarly predicted dominance of OFF segregation in 13 of
the 15 data sets (and of ON segregation in the other two data sets).
To achieve ON segregation in the majority of the sets with OFF
dominance, Lee et al. [22] introduced an inhibition term C. As ourmodel of BTDP using realistic retinal input patterns succeeded in
capturing segregation in both mouse and ferret, we suggest that therules that govern ON/OFF segregation are likely shared between
species.

were able to use the experimentally observed RGC spike trains togenerate spiking behavior in a nonlinear integrate-and-fire model;
by contrast with a reduced linear model we used only the input
correlations of the most correlated cells. The broad similarity in
our results between simulation and theory under the more relevant
BTDP plasticity rule suggests that the mechanisms for synaptic
change guided by spontaneous retinal waves are robust and do not
require the precision of individual spikes. Thus, the effect of future
manipulations of activity on segregation would not need to betested with complex spiking models, but predictions can be madewith simpler models which use spike-spike correlations over the
relevant timescales. Detailed models of retinal wave activity could
also help investigate this question, however currently they only
model the early cholinergic waves [47–49].
Unsurprisingly, we found that a synaptic plasticity rule like
STDP, which integrates spikes on much shorter millisecond-longtimescales than the relevant timescales of the input correlations,failed to segregate ON and OFF inputs. STDP usually resulted in
the strengthening of synaptic weights of both cell types, suggestinga lack of competition. STDP induces competition among synaptic
weights when the depression area of the STDP window is slightly
larger than the potentiation area (R slightly larger than one) under
independent Poisson input statistics [25]. Compared to Poisson
statistics, real RGC spike trains are highly correlated and canmore efficiently drive a postsynaptic neuron. Hence, our model
showed that physiologically unrealistic, large depression-to-potentiation ratios were necessary for segregation. When multiple retinal
spike trains were used as inputs to an integrate-and-fire model
LGN neuron, often even large ratios could not rescue segregation.
This was due to the independence in the firing of multiple ON and
OFF RGCs over the short timescales of STDP, which resulted in
nonselective (same cell type) competition. As an alternative to large
depression-to-potentiation ratios, subtractive normalization can
induce competition [27,45]. While this rescued segregation, it was
highly sensitive to parameter variation (Figure 4D,E and Table 2).
This sensitivity arises as the short integration timescales of STDP
fail to reliably sample the noisy (but approximately constant)correlation functions which extend over much longer timescales.
Due to the temporal asymmetry of STDP and the mismatch oftimescales, potentiation and depression effects cancel, reducing thesignaling component of STDP for segregation. Thus, STDP does
not utilize the correlations between cells for segregation, but rather
the cells’ time-averaged firing rates.
In addition to normalization constraints, it is possible that
segregation might result from an alternative implementation of a
spike-based plasticity rule which involves higher-order spike
integration [11–13,50]. While an exhaustive exploration of all
forms of STDP is unfeasible, one spiking plasticity rule stands out
as promising: a simple spike-coincidence rule which modifies
synaptic strength based on the number of coincident pre- andpostsynaptic spikes in a 50-millisecond time window, proposed by
Butts et al. [16]. Butts et al. [16] showed that in their highly
constrained experimental protocol of pairing one-second-longbursts to examine induced plasticity, this rule was equivalent to the
proposed BTDP. Implementing such a spike-coincidence rule inour model with the recorded RGC spike trains, however, did not
produce segregation results consistent with BTDP (data notshown), suggesting that real spike trains contain higher-order
spike dependencies that cannot be captured within a simple spikecoincidence rule.
We believe a burst-based rule is most relevant for retinogeniculate development for several reasons. It reflects the firing
patterns of RGCs during spontaneous retinal waves; by contrast,
STDP rules have been proposed for mature sensory systems where

Discussion
We have used analytical methods and computational simulations to test the hypothesis that spontaneous retinal activity guides
the segregation of ON and OFF RGCs in the developing mouse
LGN. We have compared two plasticity rules for the developmentof synaptic weights: STDP and BTDP. Modifications to these rules
adapted to the characteristics of the input firing patterns have also
been considered. Our results show that STDP alone fails tosegregate ON and OFF inputs under realistic ratios of depressionto-potentiation. STDP can segregate mixed inputs when combinedwith a homeostatic mechanism such as subtractive normalization,
however the results are highly sensitive to parameters. By
comparison, the recently-proposed BTDP rule [16] robustly
segregates ON and OFF inputs in a model LGN neuron by alocal comparison of the firing properties of neighboring RGC
inputs without requiring normalization. Although ON cells arenormally favored to win the competition because they fire before
OFF cells, our analytical work indicates that the relative
magnitude of the peak correlation of inputs influences which
population (ON or OFF) wins the competition. Hence, if OFF
RGCs are more strongly correlated than the ON RGCs, they can
win the competition. Further, our model with BTDP can also
account for segregation of inputs in the developing ferret LGN.
Our work also highlights the importance of working withexperimentally-recorded spike trains, rather than using synthetically-generated Poisson inputs, as the nature of these correlations
can strongly influence developmental outcome.
Twenty years after the initial discovery of retinal waves, there is
still an ongoing debate on if, and how, spontaneous activity
influences the development of neural connections [7,8]. The
experimental report by Kerschensteiner and Wong [20], on which
we have based our modeling, suggests that retinal waves provide
an instructive signal during the developmental age addressed. The
specific firing patterns of functionally-distinct RGCs they observed
are only present during glutamatergic waves and coincide with theperiod of ON/OFF segregation [44]. Before the switch toglutamatergic waves which occurs as circuits in the developing
retina mature, cholinergic waves with different spatio-temporalpropagation properties are believed to instruct eye-specific (lefteye/right-eye) segregation [7] (though see [8]). Theoretical models
have mostly studied the influence of cholinergic retinal waves upon
the development of neuronal connectivity in the context of the eyespecific segregation [16,45]. As far as we know, our model is thefirst to address refinements driven by the later-stage glutamatergic
waves. While eye-specific segregation in the LGN has beensuggested to be driven by summed activity over local regions in theretina [16], our prediction is that more relevant for ON/OFF
segregation in the LGN is activity between RGC pairs (as in the
reduced linear model). This is likely due to the continued
decreased convergence of retinal afferents per LGN neuron
during development [6,41,42], and receptive field development[39,46].
In this paper we have taken two complementary approaches tostudying segregation of ON and OFF inputs. With simulations, we
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ON/OFF Segregation in the LGN

single spikes can evoke postsynaptic activity [9,26], or in adeveloping retinotectal system where activity is visually-evoked[10]. Bursts are more reliable in transmitting information topostsynaptic neurons through immature synapses because of slow
and uncertain vesicle release [51]. Additionally, the developmental
switch from the NR2B to NR2A subunit composition of NMDA
receptors may be responsible for a developmental transition from asecond-long burst-based to a millisecond-long spike-based rule
governing plasticity, since NR2B has slower kinetics than NR2A
[52]. A burst-based rule like BTDP is also consistent with adetailed analysis of retinal wave characteristics, which suggested
that correlated high-frequency bursting of neighboring RGCs is
the driving factor for retinogeniculate refinements [15]. It is
possible, though, that BTDP is a phenomenological interpretationof a spike-based mechanism which can capture second-long input
correlations. To investigate such issues further, a better understanding of the biophysical mechanisms which synapses employ todetect bursts, and how they modulate synaptic strength, is needed.
To implement BTDP, we detected bursts using burst statistics
previously analyzed from the experimental recordings by Kerschensteiner and Wong [20]. Since burst statistics (firing rate,
duration, spike frequency) vary during retinal waves, and because
synapses cannot extract these statistics off-line, the biophysicalmechanisms would need to act as on-line burst detectors. Ourmodel suggests that a simple on-line spike accumulation method(Figure 2C) can detect burst latencies. Although this method will
not find the true burst onset, the error is well within the 100 ms
limit that has been suggested to preserve information content of
retinal waves [14].
In contrast to STDP with subtractive normalization, BTDP
utilized the RGC correlations over the relevant second-longtimescales to generate segregation without parameter sensitivity(for instance, to wmax ). We found that matching the integration
timescales of the synaptic plasticity rule with the timescales of
input correlations was not the only requirement to achieve robust
segregation. When we increased the timescales in STDP, we foundthat the plasticity rule always favored the cell that fired first, ON.
Satisfying both features of long timescales and temporal symmetryof input integration, BTDP successfully accounted for thedevelopment of both ON- and OFF-responsive LGN neurons inmouse [39]. Furthermore, while STDP with normalization
required careful tuning to match results from the linear andnonlinear postsynaptic models and to produce both ON- andOFF-responsive LGN neurons, BTDP demonstrated consistentsegregation irrespective of the modeling approach. With the
reduced linear model we could pinpoint how BTDP was able tointerpret RGC correlations into robust segregation. The linear
model interpretation of BTDP relates to classic models of oculardominance, where competition was induced by anti-correlated
opposite-eye activity [32]. We found that the number of ON and
OFF RGCs initially connected to the LGN neuron did not bias thesegregation outcome, unlike previously shown [22]. In fact, whichcell type won in the LGN depended on a local competition of thefiring patterns of neighboring RGCs projecting to the LGN
neuron. Additionally, the simulated integrate-and-fire model
demonstrated further loss of same-type inputs to the LGN neuron
during development: after the segregation of ON and OFF inputs(where all inputs of one type were eliminated), BTDP resulted inadditional pruning of the winning inputs to 1–3 as observed indevelopment (Figure 6A, B) [6,41].
Our model with BTDP also successfully explained ON/OFF
segregation in ferret where ON and OFF RGCs fire synchronously, but with a higher firing rate for OFF RGCs than ON [22].
Consistent with a previous covariance-based model (Text S1),
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segregation was generally in favor of the more active OFF cell. Lee
et al. [22] introduced inhibition to allow the less active ON cell towin. Despite the different input firing properties in these two
species, our model suggests that ferret and mouse have a similar
form of retinogeniculate plasticity and that the synaptic plasticity
rules which govern ON/OFF segregation are shared between
species.
In addition to segregation of RGCs onto individual LGN
neurons, neighboring LGN neurons within a sublamina in ferret
respond to the same cell type [38]. By contrast, ON- and OFFresponsive LGN neurons in mouse are distributed in a salt-andpepper pattern [39]. This difference could be due to molecular
cues [53], or more straightforwardly, lateral connectivity between
LGN neurons which we could test by modeling a population of
postsynaptic neurons. Future modeling would need to address therole of visual activity through closed eyelids together withspontaneous activity from retinal waves in ON/OFF segregationin these species. Dark-rearing in ferrets prevented ON/OFF
segregation in the LGN [54]. Our results and those of Lee et al.
[22] found that spontaneous activity is sufficient to drive ON/OFF
segregation, suggesting that visual activity may accelerate ON/
OFF segregation. Thus, it is possible that segregation would still
occur in the absence of visually-evoked activity but it would takelonger. While the effect of dark-rearing on ON/OFF segregationin the LGN has not been studied in mice, there are conflicting
reports on the development of ON/OFF segregation of RGC
dendrites [55,56]. It would be useful to manipulate spontaneous
retinal activity without manipulating visual experience, and to
examine ON/OFF segregation in dark-reared animals at laterages. This would allow for an independent evaluation of thesignificance of spontaneous and visually-evoked activity in theretina on ON/OFF segregation.
Experiments which manipulate correlations between RGCs
without eliminating spontaneous activity itself have provenextremely useful in answering the key question of whether retinal
waves influence development. Recent work reports that micelacking the b2 subunit of the acetylcholine receptor havesignificantly altered retinal wave patterns [57,58], which affectprojections in the LGN and the superior colliculus [28,39,59]. In
addition to stronger correlations between RGCs located further
apart in these mutants which are not present in wild type, one
study has found a directional bias of wave propagation in wild typeanimals [58]. The elimination of this bias in b2 mutants is
consistent with asymmetric refinement of retinal projections in
LGN and superior colliculus [28,39,59]. While these findings
represent a significant step forward towards understanding the roleof retinal activity patterns in development, along with theoreticalmodeling, they mostly address cholinergic retinal waves [16,45]. It
would be also useful to construct a more complete model for thedevelopment of the retinogeniculate pathway in two different
stages, at the retinal level capturing the varying spatio-temporal
properties of retinal waves, and at the retinogeniculate synapse. Ifcholinergic and glutamatergic waves are indeed responsible fordifferent aspects of LGN development (eye-specific and ON/OFF
segregation, respectively), these studies may provide insights into
the mechanisms by which synaptic plasticity rules interpret
ongoing changes in the firing patterns of RGCs to explain changes
in the synaptic strength of developing connections.

Materials and Methods
A synaptic weight representing the strength of the connection
between the i-th RGC and the LGN neuron is denoted by thedimensionless quantity wi , and maintained in the range ½0,wmax .
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ON/OFF Segregation in the LGN

binning the number of spikes for the i-th and k-th spike trains into
bins of size Dt~10 ms and summing the bins over the given time
lag t.
Instead of studying the evolution of all ON and OFF weights foreach data set, we studied a reduced system with a two-dimensional
weight vector w~½wON ,wOFF  (termed reduced linear model), suchthat segregation occurred whenever one weight potentiated to
wmax and the other depressed to 0. For the entries of the 2|2
correlation matrix C used in Q in Equation 8, we took

The lower bound ensures that all weights remain excitatory (in linewith Dale’s Law [60]), and the upper bound prevents the weightsfrom growing arbitrarily large [25,27]. Table 3 lists the key model
parameters used in the two different modeling approaches.
Theory: linear model. Instead of using the full set of ON
and OFF spike trains from each data set, we used pairwise spikespike correlations as inputs, and simulated postsynaptic activity
with a linear model, following the approach of Kempter et al.
[24,61]. Representing a presynaptic spike train Sipre with spikes attimes tfi as a sum of delta functions, the activity of a postsynaptic
neuron receiving inputs from all such presynaptic trains i, follows alinear Poisson model with an instantaneous time-dependent firing
rate (given as an ensemble average), SSpost T
Sipre (t)~Xd(t{tfi )"CON=ON (t)C(t)~
COFF=ON (t)

X
i

ð?
wi
{?

ð5Þ
dt’e(t{t’)SSipre T(t’),

Ch (t)~Ah e{jt{dh j=thwhere wi e(t) may be interpreted as an excitatory postsynaptic
potential (EPSP) representing the probability that the postsynaptic
neuron will fire a spike given a presynaptic spike at time t~0,
proportional to the weight wi . We used

e(t)~



K e{t=t1 {e{t=t2 ,

t§0,

0,

tv0,

ð6Þ

ð7Þ

assuming that weight modification is slower than the firing of preand postsynaptic spikes, where Q is the convolution of the input
correlation matrix C with the plasticity rule W , which we call a‘plasticity’ matrix
ð?
Qik ~
{?

ð11Þ

The parameters Ah ,dh ,th illustrated in Figure 1B were estimatedusing a nonlinear least squares procedure in R [64], and arereported in Table 1.
In the analysis of STDP, we observed a mismatch between the
results from the reduced linear and the nonlinear integrate-andfire models (Figure 4D, E). To determine the origin of thisdisagreement, we also implemented a full linear model where weused the raw correlations from Equation 9 to obtain the full
plasticity matrix Q in Equation 8. As we did not observe a
difference in the results (Table 2) between the reduced and the full
linear model with STDP, and the results of the reduced linear andthe nonlinear integrate-and-fire model with BTDP were consistent, we examined only the reduced linear model under BTDP.
Simulation: integrate-and-fire model. The retinal wave
structure of the inputs manifests itself in LGN neurons as large
periodic barrages of postsynaptic currents which drive bursts ofaction potentials [65]. To replicate realistic LGN firing patterns,
we generated postsynaptic activity of the LGN neuron according
to a nonlinear quadratic integrate-and-fire model by Izhikevich[66] (though simulating postsynaptic activity with a leaky
integrate-and-fire model [25] did not affect our results)

where t1 wt2 , t1 denotes the decay time and t2 denotes the risetimeÐ ? of the EPSP kernel, and K is a normalizing constant such that{? e(t)dt~1. Initially, we tested millisecond timescales for thesetime constants as recorded in cortical neurons [62]. We also
extended these time constants to match recordings for developing
LGN neurons [63] (Table 3), but we found no effect onsegregation outcome.
Representing the output neuron with a linear Poisson modelallows us to write a linear system for the weight dynamics
w_ ~Qw,

ð10Þ

where the correlation functions with the largest peak amplitudes
for all pairs of each cell type were selected. We fitted a symmetricdecaying exponential to the input correlation function for each
RGC pair, h~fON/ON, ON/OFF, OFF/ON, OFF/OFFg:

andf

SSpost T(t)~#CON=OFF (t),COFF=OFF (t)ð?
dsW (s) dik SSipre T(t)e(s)zdt’e(tzs{t’)Cik (t{t’) :ð8Þv’~0:04v2 z5vz140{uzwex ,ð12Þu’~a(bv{u),

ð13Þ

{?

where v and u are dimensionless variables, representing
respectively, the postsynaptic membrane potential, and amembrane recovery variable providing negative feedback to v.
When the membrane potential exceeds Vth , it is reset to c, and u isincreased by d; the variable u recovers with a timescale of a, and b
determines the resting potential (parameter values in Table 3). We
modeled the synaptic weights wi as synaptic conductances, and wex
represented the total synaptic conductance received by thepostsynaptic LGN neuron [25]. Equations were solved using aforward Euler integration scheme with a fixed time step of
dt~0:1 ms, to generate postsynaptic spikes of the same precision
as the input spikes.
Unlike the linear model, here we studied the temporal evolutionof ON and OFF weights for all inputs in each data set. To
compare results with the reduced linear model where a single ON

In particular, W denotes a functional representation of thesynaptic plasticity rule we studied, STDP or BTDP (presented
below), s is the timing between a pair of spikes of the inputs i andk, dik is the Kronecker delta equal to 1 only if i~k, and 0
otherwise, SSipre T(t) is the mean firing rate of presynaptic RGC i (:is the temporal average taken over the duration of the recording),
and Cik (t) is the input correlation function for the pair of inputs iand kCik (t)~SSipre (t) Skpre (tzt)T:ð9Þ

Note that this input correlation involves only presynaptic spiketrains, and can be derived by substituting Sipre into Spost in
Equation 5. The input correlation of Equation 9 was computed by
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ON/OFF Segregation in the LGN

For the derivation of the linear system with BTDP, we used the
spike-spike input correlations as for STDP (statistics given in
Table 1), even though BTDP modifies synaptic weights based on a
burst latency computed from the start of bursts. This was justified
because postsynaptic activity was still generated on the level of
spikes (rise in EPSP magnitude at the arrival of a presynaptic
spike), and BTDP was robust as long as the timescales of input
correlation matched the second-long integration timescales of
BTDP (discussed in Results).
To ensure smooth synaptic weight dynamics over time, we used
relatively small values of maximum potentiation and depression
amplitudes in each rule (Table 3). The resulting slow accumulationof synaptic change required a cyclic presentation of the input
spiking patterns (10–50 times) to achieve segregation of the inputs,
comparable to the 1–2 day duration of glutamatergic waves
corresponding to the period of the highest degree of retinogeniculate refinements. We studied the resulting segregation of inputs
as we varied the ratio of depression-to-potentiation, R~A{ =Az
for STDP, and R~I=Az for BTDP (even though the definitions
are different, the meaning for R is the same in both rules).
Simulation code is available as supporting material (Protocol S1).
To quantify the degree of segregation, we used the following
measure [22]and a single OFF weight were studied, we assigned all ON weights
the same initial strength, and all OFF weights the same initialstrength (but different from the initial strength for ON). Testing
different initial conditions for all 5–8 weights per data set would
have required a representation of the weight dynamics in a 5–8dimensional space. Choosing the same initial condition for all ON
and for all OFF weights, we ran a separate simulation for each
initial condition combination. Segregation of the inputs wasinterpreted as the depression of all weights of one cell type, and the
maximal potentiation of some weights of the other cell type. As we
simulated the dynamics in the full weight space with all available
spike trains in each data set, a succinct two-dimensional
representation as for the linear model was unfeasible. Note that
all ON and OFF inputs in each data were used when thatparticular data set was explored; spike trains were not mixed
between data sets. Total number of retinal afferents in each data
set were 5–8 (Table 1), which agrees with data from mouse at P12[42,67]. The data sets for both mouse and ferret are provided as
supporting material (Dataset S1) with permission from the authors.
Plasticity rules: STDP versus BTDP. According to pairbased STDP, synaptic weights were modified based on pairings
between pre- and postsynaptic spikes separated by Dt~tpost {tpre ,where the order of the spikes determined the sign of synaptic
change. We implemented STDP assuming all-to-all interactions
such that every presynaptic spike interacted with all previous
postsynaptic spikes within the integration timescale, and vice versa[25]. Each spike pair contribution was additive and independent
from all others, so the overall synaptic change was the cumulative
effect of multiple spike pairings [16,25] (though see [31] for areview on STDP). STDP (Figure 2A) was formally modeledaccording to the function

Az expð{Dt=tz Þ,
W (Dt)~{A{ expðDt=t{ Þ,

Dt§0,
Dtv0,

#ON potentiate #OFF potentiate
{
#ON
#OFF
segregation index~[½{1,1 ð16Þ#ON potentiate #OFF potentiate
z
#ON
#OFF

such that index of {1 denotes OFF segregation and index of 1
denotes ON segregation. Indices in-between indicate that the
LGN neuron responds to mixed ON and OFF inputs.
Implementation of subtractive normalization. Subtractive
normalization was implemented at the level of individual neurons
following findings in superior colliculus, where the number and thestrength of retinal afferent synapses received by a postsynaptic neuron
was preserved during development [36]. For a total of K synaptic
weights determined by the number of ON and OFF spike trains ineach data set with initial conditions in ½0,wmax , we set the total
number of weights maintained by the LGN neuron at maximal
strength wmax to be NvK. Thus, we made the postsynaptic LGN
neuron maintain a target total synaptic strength of Nwmax at all times
during simulated development. Low retinal convergence into the
LGN during adulthood has given estimates for the surviving number
of afferents, N, around 1–3 [6,41,42]. For each data set we set N to
the smaller number of total ON and OFF weights: we chose N~3 fordata set 1 and N~2 for data sets 2–6; note that for data set 6, N~2
resulted in more consistent segregation results than N~3. For
unbiased initial conditions, the subtractive normalization constraint
made all K weights start at initial strength of Nwmax =K.
Normalization was applied whenever the plasticity rule modified
synaptic strength by DS (positive if weights were potentiated, or
negative if weights were depressed) changing total synaptic strength
such that it deviated from the target Nwmax . The excess/deficit fromthis target amount was removed/added to each weight equally. For
instance, for a total of K synaptic weights each weight was updated by
DS=K. Synaptic weights were still subjected to the hard bounds
keeping them within 0 and wmax .

ð14Þ

where Az and A{ denote the maximum amount of potentiationand depression for Dt&0, and tz and t{ denote the spike
integration timescales over which synaptic potentiation anddepression are effective. Original STDP experiments report
timescales on the order of tens of milliseconds [9,10,26], which
we tested and also extended (Table 3).
In contrast to STDP, BTDP is temporally-symmetric, such thatthe timing but not the order of the bursts determines the sign ofsynaptic change over second-long timescales [16] (Figure 2B). We
implemented BTDP following the same approach and assumptions as STDP, except that in Dt~tpost {tpre , the start times of thepre- and postsynaptic bursts were used for tpre and tpost .
The function describing synaptic change according to BTDP
was experimentally fitted by Butts et al. [16] to a line with anegative (positive) slope for 0vDtƒ1 ({1§Dtw0) second, andconstant depression I for jDtjw1 second. To implement BTDP
similarly to STDP, we used an exponential representation of
BTDP which preserves the timescales and temporal symmetry of
burst integration as in the original formulation of BTDP. The
exponential approximation is given by
W (Dt)~(Az zI) exp ({jDtj=tz ){I:

ð15Þ

Here, Az denotes the maximum amount of synaptic potentiation
given a pair of precisely synchronous pre- and postsynaptic bursts,
I the amount of depression for bursts separated by more than
1 second, and tz the time constant which controls the burst
latency over which potentiation and depression occur.
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Supporting Information
A comparison between the model in Lee et al. (2002)
and BTDP.
Found at: doi:10.1371/journal.pcbi.1000618.s001 (0.06 MB PDF)

Text S1

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ON/OFF Segregation in the LGN

Table S1 Segregation results for ferret using STDP and BTDP.
Found at: doi:10.1371/journal.pcbi.1000618.s002 (0.03 MB PDF)

Acknowledgments
Daniel Kerschensteiner and Rachel Wong kindly provided the mouse data;
the ferret data came from Christopher Lee-Messer. Thanks to Jonathan
Dawes and Jean-Pascal Pfister for helpful discussions, and Keith Godfrey
for reading drafts of this paper.

Table S2 Correlation fits for ferret.

Found at: doi:10.1371/journal.pcbi.1000618.s003 (0.03 MB PDF)
Dataset S1 Spiking data from mouse and ferret.
Found at: doi:10.1371/journal.pcbi.1000618.s004 (0.70 MB ZIP)

Author Contributions

Protocol S1 C code for implementing STDP and BTDP.

Conceived and designed the experiments: JG TT SJE. Performed theexperiments: JG. Analyzed the data: JG TT SJE. Contributed reagents/
materials/analysis tools: JG SJE. Wrote the paper: JG TT SJE.

Found at: doi:10.1371/journal.pcbi.1000618.s005 (0.07 MB ZIP)

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