J Comput Neurosci (2007) 22:129–133 DOI 10.1007/s10827-006-0002-x
B RIEF COM MUNICATION
STDP rule endowed with the BCM sliding threshold accounts for hippocampal heterosynaptic plasticity Lubica Benuskova · Wickliffe C. Abraham
Received: 20 February 2006 / Revised: 13 July 2006 / Accepted: 14 July 2006 / Published online: 12 October 2006 C Springer Science + Business Media, LLC 2007
Abstract We have combined the nearest neighbour additive spike-timing-dependent plasticity (STDP) rule with the Bienenstock, Cooper and Munro (BCM) sliding modification threshold in a computational model of heterosynaptic plasticity in the hippocampal dentate gyrus. As a result we can reproduce (1) homosynaptic long-term potentiation of the tetanized input, and (2) heterosynaptic long-term depression of the untetanized input, as observed in real experiments. Keywords Heterosynaptic plasticity . Metaplasticity . STDP . Sliding BCM threshold
1 Introduction It is widely accepted that differential elevations of postsynaptic Ca2+ concentrations are crucial for the develop-
Action Editor: Nicolas Brunel L. Benuskova () Knowledge Engineering & Discovery Research Institute, Auckland University of Technology, Auckland, New Zealand e-mail:
[email protected] L. Benuskova Department of Applied Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia W. C. Abraham Department of Psychology, University of Otago, Dunedin, New Zealand e-mail:
[email protected]
ment of long-term potentiation (LTP) and long-term depression (LTD) of synaptic efficacy (Bi and Poo, 2001). Detailed phenomenological models and biophysical models of calcium-dependent synaptic plasticity that explain a wide range of experimental data have been proposed (Castellani et al., 2001; Abarbanel et al., 2002; Karmarkar and Buonomano, 2002; Shouval et al., 2002a, b; Yeung et al., 2004). All these models bring useful insights into the mechanisms of synaptic plasticity. However, they have many parameters to fit and can be computationally expensive when simulating learning in large neural networks. Therefore we are interested in developing a more abstract phenomenological model of synaptic plasticity that is computationally much less expensive than the more detailed models and yet still biologically accurate. It is known that the precise timing of pre- and postsynaptic spiking activity is important for determining LTP or LTD induction (Bi and Poo, 2001). However, several new research developments have not yet been brought into a unified theoretical framework with the STDP rules namely, the heterosynaptic plasticity and metaplasticity phenomena. Metaplasticity refers to the prior history of pre- or postsynaptic neural activity controlling the subsequent induction of synaptic plasticity at the stimulated input (Abraham and Bear, 1996). Heterosynaptic plasticity means that high-frequency stimulation (HFS, i.e., tetanization) of one set of synapses leads to synaptic plasticity in a neighbouring non-tetanized set of synapses (Abraham et al., 2001). Our goal is to bring metaplasticity, heterosynaptic plasticity and STDP into a unified theoretical framework by means of modeling heterosynaptic plasticity phenomena in the hippocampal dentate gyrus of freely moving rats (Abraham et al., 2001).
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2 Methods 2.1 Spiking model of a granule cell We constructed a simple model of a representative dentate granule cell (GC), in which we ignored the effects of inhibitory neurons and contralateral excitatory inputs. Thus, the model cell has only two inputs representing the medial and lateral perforant paths (MPP and LPP, respectively), which are two separate input pathways coming from the medial and lateral entorhinal cortices (EC), respectively, and terminating on separate but adjacent distal dendritic zones of the hippocampal dentate GCs (Fig. 1(a)) (McNaughton et al.,
Fig. 1 (a) Scheme showing our GC model inputs. (b)–(e) Evolution of MPP and LPP synaptic weights in computer simulations and real experiment expressed as % change with respect to their initial value. Empty circles always refer to experimental data (Abraham et al., 2001) and full triangles to our computer simulations. Averages of 10 runs are always depicted. In the real experiments as well as in our computer model, HFS of MPP consisted of (b) 1440, (c) 50 and (d) 10 trains of
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J Comput Neurosci (2007) 22:129–133
1981). For the neuron model we employed the simple model of a spiking neuron introduced by Izhikevich (2003). The neuron model is described by two dimensionless variables v(t) and u(t) obeying these ordinary differential equations: dv/dt = 0.04v 2 (t) + 5v(t) + 140 − u(t) + I (t)
(1)
du/dt = a(bv(t) − u(t))
(2)
The variable v(t) represents the membrane potential and u(t) represents a membrane recovery variable. After the membrane potential reaches the apex of a spike, e.g. AP = 55 mV,
ten pulses at 400 Hz (1440 M, 50 M, and 10 M, respectively). MPP HFS always starts at 30 min and the LPP HFS (50 L) always starts at 270 min. (e) Evolution of the BCM modification threshold for the three stimulation protocols. (f) Results of simulations for the 50 M + 50 L stimulation protocol for STDP without the sliding θ M . Results for both STDP implementations and for other HFS protocols were the same (data not shown)
J Comput Neurosci (2007) 22:129–133
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the membrane voltage and the recovery variable are reset according to if v(t) ≥ AP,
then
v(t) ← c u(t) ← u(t) + d
(3)
Different types of neurons have different values of dimensionless parameters a, b, c and d. We have employed the parameter values corresponding to a regularly spiking excitatory cell, i.e. a = 0.02, b = 0.2, c = − 69 mV, d = 2. Synaptic input I(t) is determined in the same way for spontaneous and evoked input activity, i.e.: I (t) = h(t) wmpp (t) Nmpp (t) + h(t) wlpp (t) Nlpp (t)
(4)
We update the neuron every 1 ms. When there is an input spike, evoked or spontaneous, at the corresponding synapse, be it LPP or MPP, function h(t) = 1 for that time moment and that pathway, otherwise h(t) = 0. Nmpp (Nlpp ) is the number of MPP (LPP) fibers, respectively, engaged by presynaptic stimulation, and w mpp (w lpp ) is the weight of the MPP (LPP) synapse, respectively. In our simulations, during spontaneous and test stimulation Nmpp = Nlpp = 150, and during HFS Nmpp or Nlpp was equal to 250. These values reflect the same ratios of stimulating intensities delivered to MPP and LPP in the real experiment (Abraham et al., 2001). The initial values of synaptic weights were chosen to be wmpp (0) = wlpp (0) ≈ 0.05, so when the two input pathways MPP and LPP were stimulated at test intensities, either simultaneously or in close temporal succession, a postsynaptic spike followed.
of 10, 50 or 1440 trains of ten pulses at 400 Hz, delivered after 30 min of test stimulation. HFS was delivered in bursts of 5–6 trains at 1-s intervals, with 30–60 s between bursts, depending on the protocol. The times required to complete the HFS were as follows: 10 trains, 1 min; 50 trains, 10 min; 1440 trains, 2 h. HFS of 50 trains was delivered to LPP at 4 h after commencement of the MPP HFS. Spontaneous input activity was generated all the time with the exception of when a given input received the trains of HFS spikes. During trains, however, we assume the non-tetanized input was still receiving the spontaneous activity, and between the bursts, we assume the two inputs had unsynchronized spontaneous activity of the frequency 8 Hz. All types of input activity were accompanied by STDP as described below. 2.3 Implementation of the STDP rule with the BCM sliding threshold To simulate synaptic plasticity and metaplasticity in our model dentate GC, we employed the STDP rule modified to incorporate the BCM sliding LTD/LTP threshold. For the STDP rule we used a formula presumably holding for hippocampal excitatory neurons, in which presynaptic spikes that precede (follow) postsynaptic spikes within a certain time window produce long-term strengthening (weakening) of synapses, respectively (Bi and Poo, 2001). Thus, the positive and negative synaptic changes, w+ (t) andw− (t), read: w+ (t) = A+ exp(−t/τ+ ) if t > 0
(5)
2.2 Simulation of presynaptic activity
w− (t) = A− exp(t/τ− ) if t < 0
(6)
Our model is solely spike-based. Input spikes along the MPP and LPP inputs were generated with specified frequencies either at random (during spontaneous activity) or periodically (during testing and HFS). There were three types of input activity in the model: (1) spontaneous activity that had a correlated and uncorrelated component, (2) test stimulation, and (3) HFS of the MPP and LPP input, respectively. Uncorrelated spontaneous spiking activity from EC via the MPP and LPP inputs was generated randomly (Poisson train) with an average frequency of τ = c(t ) exp τ −∞ τ was calculated numerically as in Benuskova et al. (2001). The averaged postsynaptic activity τ expresses the weighted average of the postsynaptic spike count, with the most recent spikes entering the sum with bigger weight than previous ones. Here, c(t) = 1, if there is a postsynaptic spike at time t, and c(t) = 0, otherwise, α is the scaling constant, and τ is the averaging period for calculation of θ M . In Abraham et al. (2001), antidromic spikes with NMDA receptors blocked were sufficient to increase the modification threshold for subsequent LTP generation, hence the rationale for the average spike count. In our implementation of STDP, the amplitudes of synaptic changes were not constant but instead they depended on the previous history of postsynaptic activity. The bigger the average postsynaptic spike count, the harder it is to achieve synaptic potentiation at all active synapses, and the easier it is to achieve synaptic depression. The opposite situation holds for low average postsynaptic activity. The values parameters used in our simulations were: A+ (0) = 0.02, A− (0) = 0.01, τ + = 20 ms, τ − = 100 ms (Froemke et al., 2005), α = 2000, and τ = 60 s. The sliding threshold, which regulates the amplitude of synaptic increase and decrease is able to prevent synaptic growth to infinity, but sometimes is not able to prevent synaptic decrease to zero. Therefore we set the minimal and maximal synaptic weight limits as being equal to 70–160% of their original value. These borders correspond to limits observed in the actual experiments (Abraham et al., 2001). 2.4 Implementations of the plain STDP rule To simulate the plain STDP, we employed Eq. (5), (6) and (7) with a synaptic decay term, i.e. w(t + 1) = w(t)(1 + w+ (t) − w − (t)) − λw(t), with λ = 0.05. The amplitudes A+ and A− were constant and equal to A+ (0) and A− (0), respectively. The second STDP implementation was the multiplicative STDP rule of the form used in Delorme et al. (2001), i.e. A+ (t) = η(w max − w(t)) and A− (t) = ηw(t), with η = 0.05 and no synaptic decay term. All other paramSpringer
Using the STDP model combined with embedded θ M , all three HFS protocols consistently evoked LTP of the MPP input, and heterosynaptic LTD of the LPP input lasting for hours (Fig. 1(b)–(d)). Another HFS of 50 trains delivered to the LPP four hours after MPP HFS commencement resulted in only partial MPP depotentiation and LPP dedepression, respectively. In the successful model, it is the postsynaptic activity that sets and maintains the value of θ M . Before the HFS of MPP, both LPP and MPP have approximately the same weights. At low-frequency largely correlated spontaneous input activity, only simultaneous or closely successive spikes at these two inputs can fire the postsynaptic GC. With the onset of selective HFS of the MPP, the activity between LPP and MPP becomes decorrelated. The HFS has a frequency of 400 Hz, and thus the train of 10 pulses lasts for 22.5 ms. At the beginning of this interval, few postsynaptic spikes are generated due to the temporal integration of the HFS activity. Later spikes are prevented due to the absolute and relative refractory periods of the GC because the firing threshold of the Izhikevich neuron depends on the history of the membrane potential before spike generation. Between the HFS bursts (intervals of 30–60 s), we assume the spontaneous activity between MPP and LPP remains decorrelated, thus leading to fewer postsynaptic spikes than before HFS. Thus, the postsynaptic spike count is decreased during the induction protocol, and so is θ M (see e.g. Fig. 1(e)). The drop in θ M enables potentiation of the tetanized MPP input, since HFS is correlated with postsynaptic spikes. However, the spontaneous input activity of the LPP input is not correlated with the GC output and as a consequence LPP weakens. After induction, in response to resumed largely correlated presynaptic spontaneous activity, the new distribution of synaptic weights leads to the same overall postsynaptic activity as before the HFS, and θ M returns to the pre-HFS value. The new weight distribution stays almost unchanged until the next HFS delivered to LPP, which generates only partial MPP depotentiation and LPP dedepression. In contrast, the plain STDP models could reproduce neither heterosynaptic LTD nor homosynaptic LTP (Fig. 1(f)).
4 Discussion STDP as originally formulated has several problems. Among others, it cannot account for the simultaneous homosynaptic LTP and heterosynaptic LTD induced by HFS of the MPP
J Comput Neurosci (2007) 22:129–133
terminating upon granule cells in the hippocampal dentate gyrus as observed in Abraham et al. (2001). We demonstrate that these experimental data can be accounted for if (1) the STDP rule is allowed to dynamically change the sizes of potentiation and depression windows according to the previous mean spike count of the postsynaptic neuron, (2) we take into account the pre- and postsynaptic spontaneous spiking activity, and (3) we decorrelate MPP and LPP spontaneous activity during the interburst episodes of the HFS, resulting in a decrease in postsynaptic spiking at these times. These assumptions of our model can be tested experimentally by manipulating either the input or postsynaptic spontaneous activities before and during HFS, and evaluating whether synaptic weight changes still occur in the same way as before. There is some experimental evidence to support our model. In the work of Tamura et al. (1992), the intraocular injection of TTX, which eliminated spontaneous activity of the retinal ganglion cells, had 2 effects: (1) it significantly increased the magnitude and induction probability of homosynaptic LTP in the visual cortex cells, and (2) heterosynaptic depression that previously accompanied the homosynaptic LTP was not observed in any of the cortices. Thus, when the presynaptic spontaneous activity was blocked there was no heterosynaptic LTD, which is in accordance with our model that explains the heterosynaptic LTD as a homosynaptic phenomenon due to presynaptic activity. We think we can safely assume that lower presynaptic activity leads to lower postsynaptic activity in their case and thus to a lower threshold for homosynaptic LTP induction. In our case it is the assumption of decorrelated presynaptic activity that leads to lower postsynaptic activity, but the effect is the same as in the study of Tamura et al. (1992), that is a lowering of the threshold for homosynaptic LTP. Another test of our model would be the investigation of whether such a modified STDP rule can lead to frequency-dependent synaptic plasticity for well-timed pre- and postsynaptic spike pairs.
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