Balancing Feed-Forward Excitation and Inhibition via Hebbian Inhibitory Synaptic Plasticity

It has been suggested that excitatory and inhibitory inputs to cortical cells are balanced, and that this balance is important for the highly irregular firing observed in the cortex. There are two hypotheses as to the origin of this balance. One assumes that it results from a stable solution of the recurrent neuronal dynamics. This model can account for a balance of steady state excitation and inhibition without fine tuning of parameters, but not for transient inputs. The second hypothesis suggests that the feed forward excitatory and inhibitory inputs to a postsynaptic cell are already balanced. This latter hypothesis thus does account for the balance of transient inputs. However, it remains unclear what mechanism underlies the fine tuning required for balancing feed forward excitatory and inhibitory inputs. Here we investigated whether inhibitory synaptic plasticity is responsible for the balance of transient feed forward excitation and inhibition. We address this issue in the framework of a model characterizing the stochastic dynamics of temporally anti-symmetric Hebbian spike timing dependent plasticity of feed forward excitatory and inhibitory synaptic inputs to a single post-synaptic cell. Our analysis shows that inhibitory Hebbian plasticity generates ‘negative feedback’ that balances excitation and inhibition, which contrasts with the ‘positive feedback’ of excitatory Hebbian synaptic plasticity. As a result, this balance may increase the sensitivity of the learning dynamics to the correlation structure of the excitatory inputs

Oscillations via Spike-Timing Dependent Plasticity in a Feed-Forward Model.

Neuronal oscillatory activity has been reported in relation to a wide range of cognitive processes including the encoding of external stimuli, attention, and learning. Although the specific role of these oscillations has yet to be determined, it is clear that neuronal oscillations are abundant in the central nervous system. This raises the question of the origin of these oscillations: are the mechanisms for generating these oscillations genetically hard-wired or can they be acquired via a learning process? Here, we study the conditions under which oscillatory activity emerges through a process of spike timing dependent plasticity (STDP) in a feed-forward architecture. First, we analyze the effect of oscillations on STDP-driven synaptic dynamics of a single synapse, and study how the parameters that characterize the STDP rule and the oscillations affect the resultant synaptic weight. Next, we analyze STDP-driven synaptic dynamics of a pre-synaptic population of neurons onto a single post-synaptic cell. The pre-synaptic neural population is assumed to be oscillating at the same frequency, albeit with different phases, such that the net activity of the pre-synaptic population is constant in time. Thus, in the homogeneous case in which all synapses are equal, the post-synaptic neuron receives constant input and hence does not oscillate. To investigate the transition to oscillatory activity, we develop a mean-field Fokker-Planck approximation of the synaptic dynamics. We analyze the conditions causing the homogeneous solution to lose its stability. The findings show that oscillatory activity appears through a mechanism of spontaneous symmetry breaking. However, in the general case the homogeneous solution is unstable, and the synaptic dynamics does not converge to a different fixed point, but rather to a limit cycle. We show how the temporal structure of the STDP rule determines the stability of the homogeneous solution and the drift velocity of the limit cycle

The effect of STDP temporal kernel structure on the learning dynamics of single excitatory and inhibitory synapses.

Spike-Timing Dependent Plasticity (STDP) is characterized by a wide range of temporal kernels. However, much of the theoretical work has focused on a specific kernel - the "temporally asymmetric Hebbian" learning rules. Previous studies linked excitatory STDP to positive feedback that can account for the emergence of response selectivity. Inhibitory plasticity was associated with negative feedback that can balance the excitatory and inhibitory inputs. Here we study the possible computational role of the temporal structure of the STDP. We represent the STDP as a superposition of two processes: potentiation and depression. This allows us to model a wide range of experimentally observed STDP kernels, from Hebbian to anti-Hebbian, by varying a single parameter. We investigate STDP dynamics of a single excitatory or inhibitory synapse in purely feed-forward architecture. We derive a mean-field-Fokker-Planck dynamics for the synaptic weight and analyze the effect of STDP structure on the fixed points of the mean field dynamics. We find a phase transition along the Hebbian to anti-Hebbian parameter from a phase that is characterized by a unimodal distribution of the synaptic weight, in which the STDP dynamics is governed by negative feedback, to a phase with positive feedback characterized by a bimodal distribution. The critical point of this transition depends on general properties of the STDP dynamics and not on the fine details. Namely, the dynamics is affected by the pre-post correlations only via a single number that quantifies its overlap with the STDP kernel. We find that by manipulating the STDP temporal kernel, negative feedback can be induced in excitatory synapses and positive feedback in inhibitory. Moreover, there is an exact symmetry between inhibitory and excitatory plasticity, i.e., for every STDP rule of inhibitory synapse there exists an STDP rule for excitatory synapse, such that their dynamics is identical

Numerical simulation of STDP-driven synaptic dynamics of a population of 120 excitatory synapses providing feed-forward input onto a single conductance based post-synaptic neuron (see e.g. Fig 3 and Methods for details).

<p>The pre-synaptic activity followed an inhomogeneous Poisson process with a time-varying intensity by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e031" target="_blank">eq (17)</a>, using <i>N</i> = 120, <i>D</i> = 10<i>Sp</i>/<i>sec</i>, <i>A</i> = 10<i>Sp</i>/sec, <i>ν</i> = 2<i>π</i>·10<i>Hz</i>. We simulated the temporally anti-symmetric exponential STDP rule (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e004" target="_blank">eq (4)</a> with <i>τ</i>_± = 20<i>ms</i>), with <i>α</i> = 1.1, <i>μ</i> = 0.1, using a learning rate constant <i>λ</i> = 5·10⁻⁴. (A and B) The synaptic weight profile <i>w</i>(<i>φ</i>, <i>t</i>) as a function of <i>φ</i> is shown for different times <i>t</i> = 0, 35,…80min by the different colors. The solid lines show the temporal average of <i>w</i>(<i>φ</i>, <i>t</i>) on the interval [<i>t</i>,<i>t</i>+1min]. (C) Traces showing the dynamics of all 120 synapses, each shown in a different color: black for the 60^th synapse (<i>φ</i>₆₀ = <i>π</i>), magenta for 120^th synapse (<i>φ</i>₁₂₀ = 2<i>π</i>), shades of blue for synapses 1–59 and shades of red for synapses 61–119. (D) The dynamics of the order parameter (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e040" target="_blank">eq (22)</a>) shown in red (x100 scaled), and the oscillation amplitude of post-synaptic activity <i>A</i>_<i>post</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e006" target="_blank">eq (6)</a>) shown in black and measured in Sp/sec, binned in time-bins of 1 minute. (E) The dynamics of the order parameter <i>ψ</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e040" target="_blank">eq (22)</a>) shown in red (calculated in time-bins of 1 sec), and the oscillation phase of post-synaptic activity <i>φ</i>_<i>post</i> (calculated in time-bins of 1 min). (F) The distribution of synaptic weights of all 120 synapses, calculated using the second half of the simulation (after convergence to a limit cycle solution). The vertical cyan line denotes the mean value of the distribution. (G) The characteristic shape of the propagating wave, calculated from synaptic weights during the second half of the simulation. Error bars depict the standard error of the mean (calculated from measurements binned at 1sec during the second half of the simulation). The calculation of <i>φ</i>_<i>post</i> was obtained using time-bins of 1 min. The vertical magenta line shows the mean phase difference between pre- and post- synaptic activity.</p

Spike Triggered Average (STA) of a single presynaptic input.

<p>The conditional mean firing rate of the presynaptic cell given that the postsynaptic cell has fired at time , is plotted as function of time. (A) Excitatory synapse (B) Inhibitory synapse. Each set of dots (color coded) is the conditional mean firing rate calculated over 1000 hours of simulation time with fixed synaptic weights and presynaptic firing rates on all inputs. The different sets correspond to a different presynaptic weight () on a single synapse on which the STA was measured. The respective dashed lines show the numerical fitting of the form where takes the revised formula: . For every type of synapse, i.e., excitatory (in A) and inhibitory (in B), the parameters describing , namely , were chosen to minimize the least square difference between the analytic expression and the numerical estimation of the STA. These parameters were then used to calculate .</p

The dynamics of the synaptic weight distribution.

<p>The probability <i>density</i> of the synaptic weight, is shown in color code as a function of time. The range of values of , , was divided into one hundred equally sized bins, and the probability of having a value in a corresponding bin of size of 1/100 was estimated numerically. The color scale is shown in terms of . The stochastic learning dynamics of a single inhibitory synapse was simulated using an integrate and fire model (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002334#s4" target="_blank">Methods</a>). The probability density was estimated from the simulations by averaging over 1999 repeats with different realizations for the noise (stochasticity of the presynaptic neurons' firing) and with initial conditions that were uniformly spaced in the interval (0, 1). Here we used , , and .</p

Spike triggered average of inhibitory presynaptic cell.

<p>The conditional mean firing rate of the inhibitory presynaptic cell given the postsynaptic cell has fired at time , is plotted as function of time, for different values for the strength of the presynaptic weight , 0.2, 0.3, 0.4, and 0.5 in red, orange, green, blue and purple circles, respectively. The dashed lines show the fits of the form with . The parameter was set to match the zero crossing point of , and we optimized the fit over the parameters and .</p

Model architecture.

<p>The STDP dynamics of a single either excitatory or inhibitory synapse is studied in purely feed-forward model. In all of the simulations presented here, the activity of the presynaptic inputs is modeled by a homogeneous Poisson process, with mean firing rate . The synaptic weights of all synapses except one is kept fixed at a value of 0.5. The post synaptic neuron is simulated using an integrate and fire model as elaborated. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#s4" target="_blank">Methods</a> for further details.</p

Numerical simulations of the STDP-driven synaptic dynamics of 1200 excitatory synapses in an isotropic “Ring Model” architecture, serving as feed-forward input onto a single conductance based post-synaptic neuron (see Methods for details).

<p>The pre-synaptic activity followed <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e031" target="_blank">eq (17)</a>, with <i>N</i> = 1200, <i>D</i> = 10<i>Sp</i>/sec, <i>A</i> = 10<i>Sp</i>/sec and a varying angular frequency <i>ν</i>. Here we simulated the temporally anti-symmetric exponential STDP rule (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e004" target="_blank">eq (4)</a> with <i>τ</i>_± = 20<i>ms</i>), with <i>α</i> = 1, <i>μ</i> = 0, using a learning rate constant <i>λ</i> = 5·10⁻³. (A) The angular velocity <i>V</i> of the order parameter <i>ψ</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e078" target="_blank">eq (34)</a> in Revolutions per Hour—RPH) is plotted as a function of the oscillation frequency. (B, C and D) The synaptic weight profile as estimated numerically (calculated from the second half of the simulations, following convergence to the limit cycle), shown as a function of the pre-post phase difference, in red. Note the weight profile and the post phase drift at the same velocity, in red. The black curve shows the zero drift solution of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004878#pcbi.1004878.e090" target="_blank">eq (40)</a>. The vertical dashed gray line depicts the order parameter or “center of mass” <i>ψ</i>₀ of the zero drift solution, the dashed blue line shows <i>ψ</i>₀ + <i>νd</i>, and the pink line shows the “center of mass” <i>ψ</i> of the weight profile of the limit cycle solution. The panels differ in terms of the oscillation frequency of the pre-synaptic population and 36 Hz for panels B, C, and D, respectively.</p

Illustration of different STDP temporal kernels () as defined by equations (7) and (8) with the “standard exponential TAH” as a reference.

<p>Each plot (normalized to a maximal value of 1 in the LTP branch) qualitatively corresponds to some experimental data. In all plots, the blue curve represents the potentiation branch , the red curve represents the depression branch and the dashed black curve represents the superposition/sum of . For simplicity, all plots were drawn with the same . (A) The “standard exponential TAH” <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Bi1" target="_blank">[1]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Froemke1" target="_blank">[18]</a>. (B) Alternate approximation to the standard exponential TAH <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Bi1" target="_blank">[1]</a>, . (C) Temporally asymmetric Anti-Hebbian STDP <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Bell1" target="_blank">[15]</a>. (D) TAH variation <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Haas1" target="_blank">[12]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Tzounopoulos1" target="_blank">[19]</a>. (E) Temporally symmetric Hebbian STDP <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Nishiyama1" target="_blank">[16]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Woodin1" target="_blank">[17]</a>. (F) Variation to a temporally asymmetric Anti-Hebbian STDP <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101109#pone.0101109-Tzounopoulos1" target="_blank">[19]</a></p