Edited by: David Hansel, University of Paris, France
Reviewed by: Carl Van Vreeswijk, CNRS, France; Maoz Shamir, Boston University, USA
*Correspondence: Lyle Muller, Unité des Neurosciences, Information et Complexité (UNIC), 1 Avenue de la Terrasse, Gif-sur-Yvette, France. e-mail:
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In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and the spiking of single neurons can be highly precise, across neuronal populations and cell types. Spike phase (i.e., the spike time with respect to a reference oscillation) is known to carry reliable information, both with phase-locking behavior and with more complex phase relationships, such as phase precession. How this precision is achieved by neuronal populations, whose membrane properties and total input may be quite heterogeneous, is nevertheless unknown. In this note, we investigate a simple mechanism for learning precise LFP-to-spike coupling in feed-forward networks – the reliable, periodic modulation of presynaptic firing rates during oscillations, coupled with spike-timing dependent plasticity. When oscillations are within the biological range (2–150 Hz), firing rates of the inputs change on a timescale highly relevant to spike-timing dependent plasticity (STDP). Through analytic and computational methods, we find points of stable phase-locking for a neuron with plastic input synapses. These points correspond to precise phase-locking behavior in the feed-forward network. The location of these points depends on the oscillation frequency of the inputs, the STDP time constants, and the balance of potentiation and de-potentiation in the STDP rule. For a given input oscillation, the balance of potentiation and de-potentiation in the STDP rule is the critical parameter that determines the phase at which an output neuron will learn to spike. These findings are robust to changes in intrinsic post-synaptic properties. Finally, we discuss implications of this mechanism for stable learning of spike-timing in the hippocampus.
In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and spiking – termed
We focused on a minimal model set up by considering a hypothetical hippocampal CA1 neuron receiving noisy, oscillating input from a large number of weak afferent inputs (see red inputs, Figure
Clearly, we expect this temporal asymmetry of STDP to create significant effects that depend on the phase of the output spike relative to the input oscillation. As an example, consider an output spike occurring on the rising phase of the inputs, where the input firing rate is increasing in time. Here, there will be more POST-PRE pairings than PRE-POST pairings, and hence we would expect a net de-potentiation (see Figure
Following these arguments, we expect STDP to structure the synaptic strengths of oscillating inputs to ensure a precise LFP-to-spike phase relationship. In fact, using analytic methods, we can calculate the location of stable phase-locking point for a neuron with STDP-modulated input synapses. We find that the location of this stable point depends on the oscillation frequency and depth of modulation of the inputs, the STDP time constants, and the ratio of de-potentiation to potentiation in the STDP rule. Because the effect of the STDP time constants and the STDP ratio – to change the relative area under each section of the piecewise exponential STDP curve – is roughly the same, in this work we consider only changing the STDP ratio. Additionally, we show that the location of this stable point is invariant to the properties of the post-synaptic neuron, such as membrane resistance or the net excitation initially applied.
For computational simplicity, the integrate-and-fire (IF) neuron model is used. The equation governing the IF neuron’s membrane potential is:
where τ
Input connections to the IF neuron are mediated by exponential synapses, without delays. When a presynaptic spike occurs, the
where τ
The 5000 presynaptic inputs to the IF neuron were modeled as inhomogeneous Poisson processes whose rate parameters oscillate in time, with a peak rate of 10 Hz. The input oscillation frequency is set to 20 Hz. To ensure that the lack of refractory period in the Poisson process inputs does not affect the results reported here, additional simulations have been performed with inputs that follow an inhomogeneous Gamma process.
The STDP rule is implemented in accordance with Song et al. (
These simulations were performed using the Brian simulator (Goodman and Brette,
We first assume an STDP rule with linear, all-to-all spike pairings. Following previous theoretical work on STDP (Song et al.,
where
The rates of the presynaptic spike trains are approximated by a continuous sinusoid (in simulations, below, we model inputs as inhomogeneous Poisson processes):
where
The output spike train is formulated as a series of delta functions:
where
To calculate the correlation function between the input oscillation and the output spike train
where
To calculate the expected weight change over time (
From Eq.
The result of the calculation (
The analytic solution has two zero crossings – one stable, the other unstable. As spike phase is inversely related to input current in the 1:1 phase-locking regime for a model neuron with a positive phase response curve (PRC), it is straightforward to see that the early fixed point will be stable, with the later unstable. For an STDP ratio of 1.05, the location of this stable fixed point falls just after 180°. By changing the STDP ratio, however, stable phase-locking can be achieved throughout a wide range of the oscillation cycle. Thus, by increasing the STDP ratio (i.e., increasing the relative amount of de-potentiation), the point of stable phase-locking will occur later in the oscillation cycle; conversely, by decreasing the STDP ratio (i.e., increasing the relative amount of potentiation), the point of stable phase-locking will occur earlier in the oscillation cycle. The change in the shape of the analytic solution with different values of the STDP ratio is illustrated in Figure
The effect of the parameter
To corroborate the analytic results, which approximate the input spike trains as a continuous sinusoid, with biophysically realistic input and output spiking processes, we performed computational simulations of an integrate-and-fire (IF) neuron and oscillating inhomogeneous Poisson process inputs. In these simulations, the IF neuron receives 5000 periodically modulated Poisson process spike inputs (see red inputs, Figure
In the control simulation without STDP (Figure
The synapses connecting the oscillating inputs to the output model neuron express STDP with linear, all-to-all spike pairings. In these simulations, all synapses are initialized to the same weight. For the first 2 s of the simulation, STDP is turned off, during the initial 1 s transient. In this way, each neuron starts out at its individual stable phase. When STDP is turned on, our putative population of output neurons quickly converges to the unique stable spike phase predicted by the analytic solution in Figure
To verify that a population of neurons will converge from outside the 1:1 phase-locking regime to our theoretically predicted stable fixed phase point, we made an additional test with an 800-neuron population receiving input from 10,000 excitatory cells. Each neuron is connected to an input with a 10% connection probability, giving an average of 1,000 input synapses per output neuron. The inputs oscillate, as above, but in this simulation, no extra DC input is added to the neurons. The simulation is run as follows. First, the simulation runs for 5 s, to integrate out any transients. Second, the simulation runs for an additional 5 s, to record the phase distribution before STDP (blue distribution, Figure
In this note, we have demonstrated through both analysis and simulation that the combination of STDP, which is classically observed in the hippocampus and neocortex (Bliss and Lomo,
This note complements and extends the work of several previous studies in stability of single neurons with plastic synapses following an STDP rule (Song et al.,
To produce the results outlined in this work, several general assumptions must hold. First, for modeling the feed-forward structures such as the hippocampal subfield CA1, we focus on the 1:1 phase-locking regime. This regime is a good approximation to the behavior of hippocampal pyramidal neurons in awake, behaving animals during spatial navigation, which fire single spikes or a closely timed 2–3 spike burst each oscillation cycle (e.g., Figure
In summary, we have considered a very simple architecture – oscillating, feed-forward input synapsing onto a single output neuron – and developed a mechanistic view of spike phase in the presence of STDP. This view provides quantitative results on spike phase during hippocampal learning, and furthermore, this same mechanism can generalize to more complex network architectures and input/output phase relationships.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors thank O. Ahmed for a careful reading of the manuscript. Lyle Muller was supported by a doctoral grant from École des Neurosciences de Paris (ENP). Romain Brette was supported by the European Research Council (ERC StG 240132). Boris Gutkin was supported by ANR, NeRF, ENP, CNRS, and INSERM.