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We selected the pointing to include both HH 30 and several stars to the south-east. A typical pointing is shown in Figure . We used roughly the same pointing each night, to minimize variations in residual flat field error. Each night we typically took two consecutive 600 second exposures in [MATH] and two consecutive 3... |
[MATH] . The image quality was typically [MATH] FWHM, and we often observed through clouds. We reduced each image by subtracting an offset calculated from the overscan, subtracting a residual bias image, and dividing by a twilight-sky flat field. We obtained instrumental magnitudes of all of the bright sources in the f... |
and an outer diameter of [MATH] . We averaged the instrumental magnitudes in the consecutive images in each filter. We adopted the star 2MASS 04314544+1814359 as a local standard. This star lies [MATH] north and [MATH] east of HH 30 and is marked in Figure . Table shows differential photometry of stars JH 192, 193, 194... |
In Table we report differential photometry of HH 30 against the local standard in the instrumental [MATH] and [MATH] systems. In Table [MATH] is the relative magnitude of HH 30 (that is, the instrumental magnitude of HH 30 minus instrumental magnitude of the local standard) and [MATH] is the standard deviation in each ... |
0.2.2 Data Set 2 Wood et al. (2000) report observations of HH 30 with Harris [MATH] filters at the 1.2 meter telescope of the F. L. Whipple Observatory on 18 nights between 1999 September 7 and 2000 February 28. HH 30 was observed more than once on 7 of these nights. These authors kindly made available their reduced ph... |
0.2.3 Data Set 3 We observed HH 30 again with the 84 centimeter telescope of the Observatorio Astronómico Nacional on Sierra San Pedro Mártir on 29 nights between 2005 September 11 and 2006 February 12. We used the POLIMA imaging polarimeter (Hiriart et al. 2005) with the SITe1 [MATH] CCD binned [MATH] and the observat... |
The POLIMA instrument has a rotating Glan-Taylor prism that serves as a polarizing filter. Each night we obtained exposures of HH 30 with the prism orientated at 0 \arcdeg , 45 \arcdeg , 90 \arcdeg , and 135 \arcdeg . We typically obtained ten 120 second exposures per night at each position during 2005 September and te... |
We reduced each image by subtracting an offset calculated from the overscan, subtracting a residual bias image, and dividing by a twilight-sky flat field. We obtained instrumental magnitudes for HH 30 using aperture photometry with an object aperture of diameter [MATH] and a sky annulus with an inner diameter of [MATH]... |
[MATH] . We averaged the instrumental magnitudes in the 0 \arcdeg and 90 \arcdeg images to produce a magnitude in the total intensity. |
We obtained an indirect photometric calibration of each night. Each night we observed the unpolarized standards Hiltner 960 and BD |
[MATH] 389 (Schmidt, Elston, & Lupie 1992). However, these standards are not photometric standards. Therefore, on three nights we observed standards from Landolt (1992) to determine the color terms for the [MATH] filter and the standard magnitudes of Hiltner 960 ( [MATH] ) and BD [MATH] 389 ( [MATH] ). We then calibrat... |
0.3 Distribution Analysis In § 0.4 we will use a null hypothesis that the data are independent and drawn from a Gaussian distribution. We will wish to use the rejection of this null hypothesis as evidence that the data are not independent. However, the data can fail this null hypothesis if they are not drawn from a Gau... |
Figure shows the distributions of the data about their means. Kolmogorov-Smirnov tests suggests that the null hypothesis that the data sets are drawn from Gaussian distributions with the same mean and standard deviation should be accepted with confidences of 0.76 (data set 1 filter [MATH] ), 0.52 (data set 1 filter [MA... |
0.4 Period Analysis Methodology 0.4.1 The Lomb-Scargle Normalized Periodogram We have investigated the presence of a periodic signal in the data using the Lomb-Scargle normalized periodogram (Lomb 1976; Scargle 1982; Press et al. 1992, §13.8). Periodic signals tend to create peaks in the periodogram. |
The data sets are characterized by separations close to multiples of 1 day and as such contain little information below the corresponding Nyquist period. Therefore, we searched for periods between 2 days and half of longest separation present in each data set (which would allow us to see two complete periods). We calcu... |
We characterized the significance of peaks in the periodogram against the null hypothesis that the data points were independent and drawn from a Gaussian distribution with mean [MATH] and variance [MATH] . We generated 10,000 trials under this null hypothesis and determined the 50%, 90%, 95%, and 99% confidence levels. |
0.4.2 Problems with Short-Term Correlations HH 30 shows short-term photometric correlations. For example, the largest intra-night peak-to-valley variability in [MATH] in data set 2 is 0.054 magnitudes (on the night of 1999 September 11), whereas the global standard deviation is 0.38 magnitudes. Less dramatically, the s... |
Short-term correlations can cause problems for period searches using the Lomb-Scargle normalized periodogram (Herbst & Wittenmyer 1996). To see this, consider a hypothetical source that varies in such a way that its magnitude over a single night is constant but the magnitude for a given night is independent of the othe... |
Consider observing this source once per night every night for 101 nights. Furthermore, consider that the observations are noiseless. The first and second panels of Figure show an example realization of this experiment and the corresponding periodogram calculated from periods of 2 days, the Nyquist period, up to 50 days... |
Now consider observing the same source twice per night, with observations separated by one hour. This generates two identical magnitudes for each night. The first and second panels of Figure show an example realization of this experiment (with the same nightly magnitudes as Figure ) and the corresponding periodogram. T... |
Thus, short-term correlations can generate peaks in the periodogram that mimic those are correlated over intervals that are short compared to the period. Thus, a periodic signal that is finely sampled will have peaks in the periodogram that arise both from short-term correlations and from the periodic signal. |
0.4.3 Mitigating Short-Term Correlations We would like to distinguish peaks caused by short-term correlations from peaks caused by periodic signals. The most rigorous solution would probably be to use a null hypothesis that incorporated the short-term correlations in the data. |
In a series of studies of stellar variability in the Orion Nebula, Stassun et al. (1999) use a null hypothesis with two Gaussians, one for intra-night variability and one for inter-night variability, Rebull (2001) uses a null hypothesis with correlated Gaussian noise, and Herbst et al. (2002) essentially modify the nul... |
In the case of HH 30 we are studying the photometric variability of a young star, but one in which almost all of the light we see is scattered by the circumstellar disk. It is not clear if the dominant variability in HH 30 is the same as in other young stars that are seen directly. Therefore, we cannot assume that the ... |
Instead, we suggest a different means to mitigate short-term correlations: we bin the data over intervals in which they are likely to be correlated if a periodic signal is present. We suggest binning the data in bins equal to a given fraction [MATH] of the period being tested. We use adaptive bins; we start the first b... |
We need to select a suitable value for [MATH] ; we have chosen 1/8 (i.e., we bin data in intervals covering 1/8 of a period). This is coarse enough to remove much of the correlations in a periodic signal but not too coarse as to completely eliminate the signal, at least for relatively smooth modulations. For data set 2... |
The third panels of Figures and show periodograms calculated after binning the data into bins of [MATH] of the period. The fourth panel in each figure shows the number of data points without binning as a dotted line and with binning as a solid line. In Figure , even though there are 202 unbinned measurements (two per n... |
In the unbinned test, the confidence level is assumed to be independent of the period. Unfortunately, in the binned test, the confidence level is now a strong function of the period being tested. To calculate the confidence levels using the standard Monte Carlo method, we assume that the probability of a false positive... |
Periodograms of binned data can still detect periodic signals. Figures and show binned and unbinned periodograms for data that are drawn from noisy periodic signals. To generate these, we added periodic component to the data used for Figure . In , the period component had a period of 5 days and peak-to-valley amplitude... |
0.5 Period Analysis Results Figures , and show the data, periodograms, and number of effective points for data sets 1, 2, and 3. The periodograms are calculated at periods ranging from 2 days to 13 days (data set 1), 87 days (data set 2), and 77 days (data set 3). As in Figures and , the first panel in each row shows t... |
The unbinned periodograms for data sets 1 and 3 show no strongly significant peaks. The highest peaks in [MATH] have periods of 12.1 and 7.5 days and significances of only slightly more than 50%. However, the unbinned periodogram for data set 2 shows peaks at periods of about 11.6, 19.9, and 69.9 days with significance... |
However, the binned periodograms for data sets 1, 2, and 3 show no significant peaks. The highest peak in [MATH] in data set 2 is still at 19.9 days but now with a significance of less than 50%. It appears that the strong peaks in the unbinned periodogram for data set 2 are entirely the result of short-term correlation... |
We mentioned above that the choice [MATH] , the bin size in units of the period being examined, is open to some debate. We used [MATH] in the figures and obtained no significant peaks in the periodogram. One might ask if other values of [MATH] might give different results. For example, in Figure , one might wonder if a... |
Recalculating the binned periodogram with [MATH] yields a peak in [MATH] in data set 2 at 19.9 days with a marginal significance of 90%. However, in order to accept this peak as indicative of a real periodic signal, we need to accept that samples of a periodic signal separated by only [MATH] of the period are still eff... |
We conclude that there is no significant evidence for a periodic photometric signal in any of the data sets. 0.6 Discussion 0.6.1 No Significant Periodic Photometric Variability |
Our analysis indicates that HH 30 shows photometric variability in [MATH] [MATH] , and [MATH] (as previously reported), but that periodograms show no significant evidence for a periodic photometric signal between periods of 2 and 87 days. This result is in disagreement with Wood et el. (2000); we suggest that correctly... |
0.6.2 Origin of the Photometric Variability The large amplitude of the variability in [MATH] (0.8, 1.1, 1.2, and 1.4 magnitudes in data sets 1, 2, and 3 and in the observations reported by Watson & Stapelfeldt (2007) along with the lack of a detected period suggests that the photometric variability in HH 30 is related ... |
0.6.3 Simultaneous HST Imaging Watson & Stapelfeldt (2007) observed HH 30 with the WFPC2 camera of the Hubble Space Telescope on 1999 February 3, coincidentally during the period in which data set 1 was obtained. At this epoch, HH 30 showed a strong lateral asymmetry in the upper nebula. The photometry of data set 1 sh... |
Acknowledgements. We thank an anonymous referee for comments which helped improve the presentation of this work. We are extremely grateful to the staff of the OAN/SPM for their technical support and warm hospitality during several long observing runs. We thank David Hiriart, Jorge Valdez, Fernando Quirós, Benjamín Garc... |
# Source: arxiv 0808.3129 # Title: Self-organization using synaptic plasticity # Sections: all # Downloaded: 2026-03-03T01:58:47.613932+00:00 |
Self-organization using synaptic plasticity Abstract Large networks of spiking neurons show abrupt changes in their collective dynamics resembling phase transitions studied in statistical physics. An example of this phenomenon is the transition from irregular, noise-driven dynamics to regular, self-sustained behavior o... |
Introduction It is accepted that neural activity self-regulates to prevent neural circuits from becoming hyper- or hypoactive by means of homeostatic processes |
. Closely related to this idea is the claim that optimal information processing in complex systems is attained at a critical point, near a transition between an ordered and an unordered regime of dynamics |
. Recently, Kinouchi and Copelli provided a realization of this claim, showing that sensitivity and dynamic range of a network are maximized at the critical point of a non-equilibrium phase transition. Their findings may explain how sensitivity over high dynamic ranges is achieved by living organisms. |
Self-Organized Criticality (SOC) has been proposed as a mechanism for neural systems which evolve naturally to a critical state without any tuning of external parameters. In a critical state, typical macroscopic quantities present structural or temporal scale-invariance. Experimental results |
show the presence of neuronal avalanches of scale-free distributed sizes and durations, thus giving evidence of SOC under suitable conditions. A possible regulation mechanism may be provided by synaptic plasticity, as proposed in |
, where synaptic depression is shown to cause the mean synaptic strengths to approach a critical value for a range of interaction parameters which grows with the system size. |
In this work we analytically derive a local synaptic rule that can drive and maintain a neural network near the critical state. According to the proposed rule, synapses are either strengthened or weakened whenever a post-synaptic neuron receives either more or less input from the population than the required to fire at... |
The model The model under consideration was introduced in and can be considered as an extension of The state of a neuron [MATH] at time [MATH] is encoded by its activation level [MATH] , which performs at discrete timesteps a random walk with positive drift towards an absorbing barrier [MATH] . This |
spontaneous evolution is modelled using a Bernoulli process with parameter [MATH] . When the threshold [MATH] is reached, the states of the other units [MATH] in the network are increased after one timestep by the synaptic efficacy [MATH] [MATH] is reset to [MATH] , and the unit [MATH] remains insensitive to incoming s... |
[EQUATION] where [MATH] is the Heaviside step function: [MATH] if [MATH] , and [MATH] otherwise. Using the mean synaptic efficacy: [MATH] we describe the degree of interaction between the units with the following characteristic parameter: |
[EQUATION] which indicates whether the spontaneous dynamics ( [MATH] ) or the message interchange mechanism ( [MATH] ) dominates the behavior of the system. As illustrated in the right raster-plot of Figure , at |
[MATH] neurons fire irregularly as independent oscillators, whereas at [MATH] (central raster-plot) they synchronize into several phase-locked clusters. The lower [MATH] , the less clusters can be observed. For [MATH] |
the network is fully synchronized (left raster-plot). In it is shown that the system undergoes a phase transition around the critical value [MATH] . The study provides upper [MATH] ) and lower bounds ( [MATH] ) for the mean inter-spike-interval (ISI) [MATH] of the ensemble and shows that the range of possible ISIs taki... |
for a similar neural model. The average of the mean ISI [MATH] is of order [MATH] with exponent [MATH] for [MATH] [MATH] for [MATH] , and [MATH] for [MATH] |
as [MATH] , and can be approximated as shown in with [EQUATION] Self-organization using synaptic plasticity We now introduce synaptic dynamics in the model. We first present the |
dissipated spontaneous evolution , a magnitude also maximized at [MATH] . The gradient of this magnitude turns to be simple analytically and leads to a plasticity rule that can be expressed using only local information encoded in the post-synaptic unit. |
3.1 The dissipated spontaneous evolution During one ISI, we distinguish between the spontaneous evolution carried out by a unit and the actual spontaneous evolution needed for a unit to reach the threshold [MATH] . The difference of both quantities can be regarded as a surplus of spontaneous evolution, which is dissipa... |
Figure a shows an example trajectory of a neuron’s state. First, we calculate the spontaneous evolution of the given unit during one ISI, which it is just its number of stochastic state transitions during an ISI of length [MATH] (thick black lines in Figure a). These state transitions occur with probability [MATH] at e... |
[MATH] over many spikes and all units we can calculate the average total spontaneous evolution: [EQUATION] Since the state of a given unit can exceed the threshold because of the received messages from the rest of the population (blue dashed lines in Figure |
a), a fraction of ( ) is actually not required to induce a spike in that unit, and therefore is dissipated. We can obtain this fraction by subtracting from ( ) the actual number of state transitions that was required to reach the threshold [MATH] . The latter quantity can be referred to as effective spontaneous evoluti... |
and is on average [MATH] minus [MATH] , the mean evolution caused by the messages received from the rest of the units during an ISI. For |
[MATH] , the activity is self-sustained and the messages from other units are enough to drive a unit above the threshold. In this case, all the spontaneous evolution is dissipated and [MATH] . Summarizing, we have that: |
[EQUATION] If we subtract ( ) from [MATH] ), we obtain the mean dissipated spontaneous evolution, which is visualized as red dimensioning in Figure a: |
[EQUATION] Using ( ) as an approximation of [MATH] we can get an analytic expression for [MATH] Figures b and c show this analytic curve [MATH] in function of [MATH] together with the outcome of simulations. |
At [MATH] the units reach the threshold [MATH] mainly because of their spontaneous evolution. Hence, [MATH] and [MATH] . The difference between [MATH] and [MATH] increases as [MATH] |
approaches [MATH] because the message interchange progressively dominates the dynamics. At [MATH] , we have [MATH] . In this scenario [MATH] , is mainly determined by the ISI [MATH] and thus decays again for decreasing [MATH] . The maximum can be found at [MATH] |
3.2 Synaptic dynamics After having presented our magnitude of interest we now derive a plasticity rule in the model. Our approach essentially assumes that updates of the individual synapses [MATH] are made in the direction of the gradient of [MATH] . The analytical results are rather simple and allow a clear interpreta... |
We start approximating the terms [MATH] and [MATH] by the sum of all pre-synaptic efficacies [MATH] [EQUATION] This can be done for large [MATH] and if we suppose that the distribution of |
[MATH] is the same for all [MATH] [MATH] is now defined in terms of each individual neuron [MATH] as: [EQUATION] An update of [MATH] occurs when a spike from the pre-synaptic unit [MATH] |
induces a spike in a post-synaptic unit [MATH] . Other schedulings are also possible. The results are robust as long as synaptic updates are produced at the spike-time of the post-synaptic neuron. |
[EQUATION] where the constant [MATH] scales the amount of change in the synapse. We can write the gradient as: [EQUATION] For a plasticity rule to be biologically plausible it must be local, so only information encoded in the states of the pre-synaptic [MATH] and the post-synaptic |
[MATH] neurons must be considered to update [MATH] We propagate [MATH] to the state of the post-synaptic unit [MATH] by considering for every unit [MATH] , an effective threshold [MATH] which decreases deterministically every time an incoming pulse is received |
. At the end of an ISI [MATH] and encodes implicitly all pre-synaptic efficacies of [MATH] Intuitively, [MATH] indicates how the activity received from the population in the last ISI differs from the activity required to induce and spike in [MATH] |
The only term involving non-local information in ( 10 ) is the noise rate [MATH] . We replace it by a constant [MATH] and show later its limited influence on the synaptic rule. With these modifications we can write the derivative of [MATH] with respect to [MATH] as a function of only local terms: |
[EQUATION] Note that, although the derivation based on the surplus spontaneous evolution 10 ) may involve information not locally accessible to the neuron, the derived rule ( 11 ) only requires a mechanism to keep track of the difference between the natural ISI and the actual one. |
We can understand the mechanism involved in a particular synaptic update by analyzing in detail Eq. ( 11 ). In the case of a negative effective threshold ( [MATH] ) unit [MATH] receives more input from the rest of the units than the required to spike, which translates into a weakening of the synapse. Conversely, if [MA... |
for practical purposes. Figure a shows Eq. ( 11 ) in bold lines together with [MATH] (dashed line, corresponding to [MATH] ) and |
[MATH] (dashed-dotted, [MATH] ), for different values of the effective threshold [MATH] of a given unit at the end on an ISI. [MATH] indicates the amount of synaptic change and |
[MATH] determines whether the synapse is strengthened or weakened. The largest updates occur in the transition from a positive to a negative [MATH] and tend to zero for larger absolute values of [MATH] Therefore, significant updates correspond to those synapses with post-synaptic neurons which during the last ISI have ... |
We remark the similarity between Figure b and the rule characterizing spike time dependent plasticity (STDP) Although in STDP the change in the synaptic conductances is determined by the relative spike timing of the pre-synaptic neuron and the post-synaptic neuron and here it is determined by [MATH] at the spiking time... |
Figure b illustrates the role of [MATH] in the plasticity rule. For small [MATH] , updates are only significant in a tiny range of [MATH] values near zero. For higher values of [MATH] , the interval of relevant updates is widened. The shape of the rule, however, is preserved, and the role of [MATH] is just to scale the... |
[MATH] 3.3 Simulations In this section we show empirical results for the proposed plasticity rule. We focus our analysis on the time |
[MATH] required for the system to converge toward the critical point. In particular, we analyze how [MATH] depends on the starting initial configuration and on the constant [MATH] |
For the experiments we use a network composed of [MATH] units with homogeneous [MATH] and [MATH] . Synapses are initialized homogeneously and random initial states are chosen for all units in each trial. Every time a unit [MATH] fires, we update its afferent synapses [MATH] , for all [MATH] , which breaks the homogenei... |
The network starts with a certain initial condition [MATH] and evolves according to its original discrete dynamics, Eq. ( ), together with plasticity rule ( ). To measure the time |
[MATH] necessary to reach a value close to [MATH] for the first time, we select a neuron [MATH] randomly and compute [MATH] every time [MATH] fires. We assume convergence when [MATH] for the first time. In these initial experiments, [MATH] is set to [MATH] and [MATH] is either [MATH] or |
[MATH] We performed [MATH] random experiments for different initial configurations. In all cases, after an initial transient, the network settles close to [MATH] presenting some fluctuations. These fluctuations did not grow even after |
[MATH] ISIs in all realizations. Figure shows examples for [MATH] We can see that for larger updates of the synapses ( [MATH] ) the network converges faster. However, fluctuations around the reached state, slightly above [MATH] , are approximately one order of magnitude bigger than for |
[MATH] We therefore can conclude that [MATH] determines the speed of convergence and the quality and stability of the dynamics at the critical state: high values of [MATH] cause fast convergence but turn the dynamics of the network less stable at the critical state. |
We study now how [MATH] depends on [MATH] in more detail. Given [MATH] and [MATH] , we can approximate the global change in [MATH] |
after one entire ISI of a random unit assuming that all neurons change its afferent synapses uniformly. This gives us a recursive definition for the sequence of [MATH] |
[EQUATION] [EQUATION] Then the number of ISIs and the number of timesteps can be obtained by [EQUATION] Figure shows empirical values of [MATH] and |
[MATH] for several values of [MATH] together with the approximations ( 12 ). Despite the inhomogeneous coupling strengths, the analytical approximations (continuous lines) of the experiments (circles) are quite accurate. Typically, for [MATH] more spikes are required for convergence than for [MATH] . However, the oppos... |
) present in the system if [MATH] , causes the system to be more resistant against synaptic changes, which increases the number of updates (spikes) necessary to achieve the same effect as for [MATH] . Nevertheless, since the ISIs are much shorter for supercritical coupling the actual number of time steps is still lower... |
Discussion Based on the amount of spontaneous evolution which is dissipated during an ISI, we have derived a local synaptic mechanism which causes a network of spiking neurons to self-organize near a critical state. Our motivation differs from those of similar studies, for instance |
, where the average branching ratio [MATH] of the network is used to characterize criticality. Briefly, [MATH] is defined as the average number of excitations created in the next time step by a spike of a given neuron. |
The inverse of [MATH] plays the role of the branching ratio [MATH] in our model. If we initialize the units uniformly in [MATH] , we have approximately one unit in every subinterval of length [MATH] , and in consequence, the closest unit to the threshold spikes in [MATH] cases if it receives a spike. For [MATH] , a spi... |
[MATH] . Conversely, for [MATH] , the spike of a single neuron triggers more than one neuron to spike ( [MATH] ). Only for [MATH] the spike of a neuron elicits the order of one spike ( [MATH] ). Our study thus represents a realization of a local synaptic mechanism which induces global homeostasis towards an optimal bra... |
This idea is also related to the SOC rule proposed in where a mechanism is defined for threshold gates (binary units) in terms of bit flip probabilities instead of spiking neurons. As in our model, criticality is achieved via synaptic scaling, where each neuron adjusts its synaptic input according to an effective thres... |
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