There is a growing interest in developing novel brain stimulation methods to control disease-related aberrant neural activity and to address basic neuroscience questions. Conventional methods for manipulating brain activity rely on open-loop approaches that usually lead to excessive stimulation and, crucially, do not restore the original computations performed by the network. Thus, they are often accompanied by undesired side-effects. Here, we introduce delayed feedback control (DFC), a conceptually simple but effective method, to control pathological oscillations in spiking neural networks (SNNs). Using mathematical analysis and numerical simulations we show that DFC can restore a wide range of aberrant network dynamics either by suppressing or enhancing synchronous irregular activity. Importantly, DFC, besides steering the system back to a healthy state, also recovers the computations performed by the underlying network. Finally, using our theory we identify the role of single neuron and synapse properties in determining the stability of the closed-loop system.
Open-loop brain stimulation has emerged as a common tool to restore aberrant neuronal activity. The most successful example is the application of high-frequency deep-brain-stimulation (DBS) used to ameliorate motor symptoms in Parkinson’s disease (PD). However, even in this case the stimulation induces side-effects such as gait imbalance, cognitive impairment, speech impairment, depression etc. [1]. The main cause of these side-effects is likely to be the constant stimulation, but additional explanations are plausible, e. g. the inability of open-loop stimulation to recover the original computations carried out by the impaired brain area. Thus, there is a clear need for more sophisticated brain stimulation schemes [2–4]. Moreover, to exploit the full potential of external brain stimulation as a research and therapeutic tool it is important to obtain theoretical insights that can guide the design of novel stimulation protocols. The goal for these new stimulation methods should ideally be twofold: to alter the dynamical state of the brain activity in a desired manner and to recover the computations performed by the network. Here, we demonstrate that DFC, a method with its origin in chaos control [5,6], can achieve these objectives. To show that DFC is effective in altering the global activity state, we focus on its ability to switch the network state between synchronous-irregular (SI) oscillatory and asynchronous-irregular (AI) non-oscillatory activity. This choice is motivated by the fact that several brain diseases are manifested as a transformation of the AI state to persistent SI oscillations, e. g. in PD [7] and in certain forms of epilepsy [8], or as the inability of the network to generate transient SI activity, e. g. in schizophrenia [9]. To demonstrate that DFC facilitates the recovery of certain types of computations, we also illustrate how a network under DFC can effectively process and route rate as well as temporally coded signals. Thus, DFC not only steers the system to a more physiological activity regime, but it also recovers to a considerable degree the coding abilities of the network as they were present before the onset of the pathology. Previous theoretical models of closed-loop stimulation are not suitable to study the control of SI oscillations because the dynamics that arise in networks of phase oscillators [10–12], in networks of Hodgkin-Huxley neurons [13,14] and in Wilson-Cowan type firing rates models [15] are qualitatively different from the SI oscillations [16,17]. In addition, the physiologically plausible SI oscillations are known to be robust to both noise and heterogeneities [18–20] and, therefore, require a more differentiated control approach. Our control strategy is applicable to any network with arbitrary connectivity that undergoes a Hopf bifurcation and it is useful but not critical to know the parameters a priori. In fact, we show how adaptive tuning methods can be used to estimate the control parameters if the precise network parameters are not known. The DFC based stimulation method proposed here can be in principle applied in all animal models where the use of optogenetic tools allows for direct modulation of the membrane potential. In human patients, where optogenetics is currently not an option, manipulation of the membrane potential can be achieved indirectly via subthreshold electrical stimulation. Importantly, the method is not restricted to stimulation of deep structures, but could be applied non-invasively to modulate activity in cortical layers as well [21]. Finally, the theoretical insights we provide into the mechanisms of feedback control in SNNs could also explain the recent success of event-driven stimulation schemes [22–24].
While our goal is to reveal the mechanisms by which DFC controls SI activity in excitatory-inhibitory SNNs, it is more instructive to first demonstrate the concept in a simple, purely inhibitory SNN. In the AI state the population average of the firing rate is constant in time r (t) = r0. Therefore, the mean recurrent input that each neuron receives is also constant: I r e c (t) = C · J / C · (s ⋆ r) [ t ] = C · J / C · ∫ s (τ) r (t - τ) d τ = J · r 0 · ∫ s (τ) d τ where C is the average in-degree, J/C the synaptic coupling strength and s (t) the postsynaptic current. In such a network, the emergence of SI oscillations can be investigated by analyzing the stability of the network firing rate in the AI state [18,19]. A small perturbation in the steady-state firing rate r (t) = r 0 + R e [ r ^ 1 (λ) e λ t ] where eλt is an eigenmode of the network dynamics with complex eigenvalue λ, leads to a perturbation in the recurrent input I (t) = I 0 + R e [ I ^ 1 (λ) e λ t ] with I ^ 1 (λ) = - J S (λ) r ^ 1, where J is the synaptic coupling strength and S is the synaptic response function. In a recurrent network both perturbations have to be consistent, that is r ^ 1 (λ) = R (λ) I ^ 1 (λ) where R is the neuron response function. This results in the self-consistency equation: J · R (λ) · S (λ) = 1 (1) In a purely inhibitory network J is negative, but here the negative sign of J has been absorbed in the phase S (λ). We can then compute the eigenvalue spectrum, that is the roots λ that satisfy Eq (1). When the eigenvalues have a positive real part, the AI state is unstable and the SNN settles in the SI state. Note that due to the synaptic delays the spectrum is infinite. However, in time-delay systems of the retarded type that we are considering here, the total number of unstable eigenvalues is always finite [25]. Increasing J shifts the spectrum towards more positive values on the real axis. For a critical value Jcr a complex pair of eigenvalues crosses the imaginary axis and the system becomes unstable through a supercritical Hopf bifurcation [18] (Fig 1). In the following we consider a SNN in which J > Jcr, thus resulting in the emergence of SI oscillations. We aim at designing a controller that can alter the global activity state from SI to AI by placing the unstable eigenvalues back to the left half-plane (Fig 1e). For the implementation of our DFC controller we made two assumptions. First, we chose the instantaneous population rate to be the output state of the system. This state needs to be continuously monitored by the controller. Based on this state, the controller evaluates an appropriate control signal. The second assumption is that the control signal can be applied via current injection directly to the somata of the neurons. Note that the control signal is identical for all neurons in the network. Thus, the synchrony in the network activity was not decorrelated because each neuron received different input. To include the contribution of DFC the self-consistency Eq (1) needs to be modified: J R (λ) S (λ) e - λ · d - K R (λ) M (λ) e - λ · d c = 1 (2) where K is the control gain, dc the control delay and M the control kernel. The roots of the above equation (see methods) yield the range of parameters K, dc that move the unstable eigenvalues back to the left-half plane (Fig 1e), which results in a switch of activity from SI to AI. To verify the analytical solution we simulated an Erdös-Rényi type inhibitory recurrent network of N = 10,000 sparsely connected leaky-integrate-and-fire (LIF) neurons. The neurons were connected with a connection probability ϵ = 0. 1. Switching on the DFC with parameters estimated from Eq (2), almost immediately results in suppression of oscillations and in a network state that resembles the AI regime (Fig 2a–2d). The suppression of stochastic oscillations is evident both in the spiking activity of single neurons (Fig 2a) and in the population activity of the network (Fig 2b). The spike count variability and the irregularity of single neuron interspike intervals, estimated by the Fano Factor (FF) and the coefficient of variation (CV) respectively, confirm that under DFC the firing of individual neurons in the network follows Poisson statistics (AI: FF = 1. 04, CV = 1. 01, DFC: FF = 1. 02, CV = 0. 99). Moreover, the oscillation index that captures the degree of oscillatory activity (see methods) is in both conditions comparable (AI: PT = 1. 47, DFC: PT = 1. 45) and significantly smaller than in the SI state (PT = 3). The change in the network spiking activity is also observed in the subthreshold membrane potential of individual neurons (Fig 2c and 2d). Next we demonstrate the applicability of DFC in changing the SI state in recurrent networks of excitatory and inhibitory neurons. To this end we simulated a SNN composed of 8000 excitatory and 2000 inhibitory neurons with Erdös-Rényi type connectivity and a connection probability of ϵ = 0. 1. As in the I-I network, we used the mean-field approach to derive appropriate values for the parameters to attain an SI state. The self-consistency equation for the coupled EI-network under DFC is given by [ (J E I · S E I (λ) e - λ · d E I - K · M (λ) e - λ · d c) · R E (λ) ] · J I E · R I E (λ) · S I E (λ) · e - λ · d I E = 1 (3) where Jij, dij is the synaptic coupling strength and delay from population j to population i and RE (RI) is the neuron response function of excitatory (inhibitory) neurons. Note that we ignored the recurrent couplings within E and I populations, because we were interested only in the oscillations created by the EI-loop. We implemented DFC by recording the activity of neurons in the inhibitory population while stimulating excitatory neurons. Here again K is the control gain, dc the control delay and M the control kernel. Switching on the controller yielded a near instantaneous transition in the network activity from SI to AI (Fig 2e–2h). In this case the original physiological state that we wanted to recover was characterized by slightly less irregular firing of the individual neurons. Nevertheless, DFC successfully steered the network to a regime with statistics comparable to the AI activity (DFC: FFE = 0. 85, CVE = 0. 91, AI: FFE = 0. 83, CVE = 1. 03). In a coupled network with more than one population additional possibilities for recording and stimulating neurons exist. For instance, we could both record and stimulate the excitatory population (see below “stability and robustness of control domains”). Our results, however, do not depend on the exact identity of the recorded and stimulated neurons. The oscillation frequency in the SI state of the network was in the beta range, f ≈ 25–30Hz, which is characteristic for PD [7]. This suggests that if PD-associated beta oscillations are caused by a strong coupling between STN and GPe [26,27], then our DFC approach could be used to suppress these beta band oscillations. The results presented here are general and the same approach can be applied to suppress oscillations in other frequency bands as well as long as the oscillations underlie a Hopf bifurcation. To determine the range of values that led to stable control we fixed the control kernel M, using a box function of width 1 ms, and parametrized the system by the control gain K and delay dc. For each pair of values we simulated the SNN and computed the oscillation index (Fig 3a and 3c). The (K, dc) -plane shows that a stable control domain exists at 7 ms. That is, an effective control delay of dc, eff = 7 ms yields the maximum stability for the resulting AI state. The semi-analytical results from mean-field theory (red contour lines derived from Eqs 2 and 3 for the I-I network and E-I network, respectively) are in good agreement with the numerical simulations (blue shades). The only discrepancy occurs when the difference between synaptic and control coupling is small. In such a scenario it is more difficult to maintain constant rates of the stimulated population and the system may become effectively excitatory leading to rate instabilities [28]. Moreover, fluctuations in the mean input that are ignored in our mean-field approach could also become more important. The analysis of these fluctuations is beyond the scope of this work and will be addressed in a future study. Despite the fact that one stable control domain exists, a compensation mechanism to maintain constant firing rates is required to achieve stable control. In real-life applications a detailed fine-tuning may not always be possible. Therefore, we modified our control protocol and introduced an additional delay term dc2, thus, effectively feeding into the controller the difference between two time-delayed versions of the population activity. For such differential DFC scheme the control signal is given by (see methods): I C (t) = K · M (t) ⋆ (v (t - d c 1) - v (t - d c 2) ) Differential control has been previously used to control unstable periodic orbits [5,29] and to suppress synchrony in networks with discrete-time neuron models [30]. With the differential control we accounted for the fact that recording neural activity and injecting a control current into the neurons introduces a finite time-delay. Therefore, we used a small but non-zero value for dc2, i. e. dc2 = 1 ms, which is close to the overall closed-loop delay introduced by current technologies [31,32]. A crucial advantage of differential DFC is that no additional rate compensation is required, because the mean contribution of the control signal vanishes l i m T → ∞ 1 T ∫ 0 T (v (t - d c 1) - v (t - d c 2) ) d t = 0 Moving in the control parameter space (K, dc), therefore, did not affect the firing rates of the neurons. This was reflected in the near perfect overlap of theoretical predictions and numerical simulations of the SNN (Fig 3b). In addition, differential DFC introduced two positive effects on the stability of the control domains: (i) The first control domain was expanded, which amounts to an increase in the robustness in the parameter variation. That is, small deviations from the estimated values of the gain and the delay would not be critical for the stability of the AI state achieved by differential control. (ii) A new stable control domain appeared at t = 23 ms. Thus, with differential control there is an increase of the range of parameters that lead to stability. DFC also enhanced the robustness of the system to external disturbances, e. g. undesired signals at the controller output, measurement noise etc. This becomes evident when we consider the distance Bcr of the complex eigenvalues λi from the imaginary axis for the main stable control domain at t = 7 ms. A more robust closed-loop system is reflected in higher values of Bcr. Differential and direct control yielded B d i f f c r = m a x (R e (λ i) ) = - 257 and B d i r e c t c r = m a x (R e (λ i) ) = - 224, respectively, clearly revealing a more robust system with differential DFC. Both direct and differential control were effective in a E-I network as well. The location of the stable control domains depended on the exact implementation (Fig 3c–3e). When the activity of the inhibitory population was monitored while stimulating the excitatory population, the main stable control domain appeared at t = 7 ms (Fig 3c). This location is identical with the purely inhibitory network and reflects the overall delay of the I-E path (I-I loop) in the E-I (I-I) network. Indeed for both the I-E path and I-I loop the effective delay is deff = 7 ms (see methods). By contrast, when the excitatory population was both recorded and stimulated then the location of the domains shifted to around t = 14 ms reflecting the larger overall delay in the E-I-E loop. (Fig 3d and 3e). Note that in this case the stable control domain for direct control was smaller. The reason is that the size of the stable control domains shrinks for larger delays. In both the I-I and E-I SNNs we applied an identical control signal to all stimulated neurons. That is, we did not disrupt oscillations and decorrelated network activity by injecting different currents to each of the neurons. This is in contrast with a widespread assumption that common input always tends to increase correlations in neural activity [33]. The results from the application of DFC reveal that common input can both increase or decrease correlations in SNNs. It is the timing and the amplitude of the common input that determines the direction in which correlations are affected. It is important to point out that injection of a control signal is not equivalent to the application of additive noise to the system. To demonstrate this we simulated an I-I network and injected Gaussian noise with the same mean and variance as the control signal to all neurons. This stimulation approach failed to suppress SI oscillations (Fig 4a and 4b) indicating that the temporal structure of the control signal is crucial for successful control. Increasing further the noise intensity, e. g. by a factor of ten, eventually resulted in desynchronization of the activity and in quenching of oscillations (Fig 4d and 4e). However, with such strong external noise the network dynamics is predominantly influenced by the input rather than the recurrent activity. This condition is disastrous from a computational point of view, because any information processing taking place within the stimulated brain region would be severely impaired. To illustrate this we recorded the subthreshold dynamics of ten randomly selected neurons in the network (Fig 4f). The huge fluctuations in the membrane potential under the influence of strong external noise are rather pathological. By contrast, the fluctuations in the case of DFC are comparable to those in the physiological AI regime. The detrimental effect of strong external noise became even more apparent when we studied the response of the network to incoming stimuli. We examined two scenarios. First, we tested how a series of incoming pulse packets composed of randomly distributed spikes are processed by the SNN. We evaluated the network response by the area under the curve (AUC, see methods) for each of the following network states: AI, SI quenched DFC and SI quenched by noise stimulation. A high AUC value reflects better separability of two conditions. It is evident that the AUC in the AI state and in the DFC condition is close to unity indicating that both conditions are comparable in terms of stimulus separability (Fig 5a and 5b). By contrast, when the SI oscillations were quenched by the injection of strong external noise the AUC dropped significantly. That is, DFC, in contrast to strong noise stimulation, does not impair the ability of the network to detect incoming stimuli. Next we tested how DFC affects temporal aspects of the network response. To this end, we provided external correlated inputs to all stimulated neurons and measured the spike train similarity in the network response. We computed the spike distance D that captures the time-resolved degree of synchrony between individual spike-trains ([34], see methods). Again DFC did not impair the temporal processing as indicated by a clear separation of the two clusters during baseline DB and stimulation DS (Fig 5c). For external noise, however, the two distributions of values strongly overlapped, showing that aspects of temporal processing as measured by pairwise synchrony are clearly compromised when the SI state is disrupted by open-loop noise injection. These results suggest that processing of incoming signals either locally or by downstream areas is feasible in a DFC scheme, but not in an external noise scheme. The above two results clearly demonstrate that DFC has multiple advantages compared to the open-loop noisy stimulation. DFC does not only suppress SI activity steering the network to an AI regime, it also facilitates the recovery of the network’s ability to process stimulus related information. From its design it is evident that DFC effectively counteracts the increase in coupling strength, which is one of the main causes for the emergence of SI activity. Indeed, the goal of the DFC design was to move the poles of the system at, or close, to their original positions. Ideally, the stimulation kernel M would match the synaptic kernel S with dc = d and the amplitude of the control gain K would be tuned to match the pathological increase of the coupling strength ΔJ. If this were the case, DFC would completely eliminate the effects on the mean recurrent input. This is evident if we consider the modulation to a perturbation in the average input to a neuron I (λ) = (J + Δ J) · R (λ) · S (λ) · e - λ d - K · R (λ) · M (λ) · e - λ d c = K = Δ J, M = S (J + Δ J) · R (λ) · S (λ) · e - λ d - Δ J · R (λ) · S (λ) · e - λ d = J · R (λ) · S (λ) · e - λ d That is, under DFC the effects of ΔJ are not visible in the perturbed current term. In practical applications, of course, a perfect match between the control parameters (K, dc, M) with the synaptic values is not feasible, because the exact shape of the synaptic kernels are not known a priori and have to be estimated. Nevertheless, within a certain reasonable range of parameters (see also section “stable control domains”), DFC still places the eigenvalues close to the initial position they had before the onset of pathology. Therefore, as we showed above, aspects of both rate and temporal coding that the network may be performing are recovered. The understanding of the exact mechanisms by which DFC suppressed SI activity allowed us to precisely investigate how the neuron and synapse response function R and S respectively influence the stability of the closed-loop system. To this end, we again used a mean-field approximation, which explicitly incorporates the expressions for R and S. In general, the neuron response R depends on the specific neuron model as well as on the external input. Here, we did not change the neuron model, but altered the external Gaussian white noise input by using different values for the mean and variance (μ, σ2). We then assessed the stability of the system. It is apparent that for a given pair of coupling and control parameters (J, d) and (K, dc), respectively, the system becomes unstable as we move in the two dimensional parameter-space spanned by the mean and variance (Fig 6a). For meaningful comparison we used (μ, σ2) -combinations that yield constant rates. In the ideal case where M (λ) = S (λ) and dc = d Eq (2) becomes: J · R (λ) · S (λ) · e - λ d - K · R (λ) · M (λ) · e - λ d c = 1 (J - K) · R (λ) · S (λ) · e - λ d = 1 (J - K) · G S · R n (λ) · S (λ) = 1 (4) where Gs is the slope (or the static gain) of the ‘f-I curve’ at the operating point and Rn the normalized neuron response (see methods). The critical effective coupling is then given by L c r (λ) = (J - K) (λ) = 1 G s | R n (λ) · S (λ) | As we move along the constant output firing rate lines both Gs and |Rn (λ) | increase (S1b and S1b Fig) leading to a decrease of Lcr. The changes in Gs are significantly larger than those in |Rn (λ) |, implying that the static gain is the dominant factor that affects stability. The changes in |S (λ) | are negligible (S1c Fig). This is expected, because the frequency range we are interested in is much smaller than the cut-off frequency of the synaptic filter ω < ω3db. Thus, when the system operates in a dynamic regime in which single neuron responses have a higher gain the control domains shrink and the range of K values that stabilizes the system decreases. Next, we investigated the interaction between the synaptic S (λ) and the control kernel M (λ). The amplitude responses for different kernels do not vary significantly (S2 Fig). Therefore, the important factor that influences stability is the phase difference or, alternatively, the difference Δd between the effective delays of the synaptic deff and the coupling kernel dc, eff. An optimal result is achieved if this difference vanishes (see methods) i. e. when Δ d = d e f f - d c, e f f = 0 This point is illustrated for the case where dc = d+1ms (Fig 6c). These results show that DFC does not strongly depend on the shape but rather on the effective delay of the kernel. Interestingly, the same control strategy can be used to induce or enhance rather than to suppress oscillations. Choosing appropriate control parameters to increase the effective coupling, i. e. selecting K to have the same sign as J (see methods), results in SI activity (Fig 7). This may be helpful for the treatment of symptoms in several pathological conditions that are characterized by impaired oscillations, e. g. gamma power decrease in schizophrenia [35]. Thus, DFC is a generic control approach and the control parameters (K, dc, M) can be tuned to quench or to enhance oscillatory activity, depending on the nature of aberrant activity.
Few studies have addressed the problem of suppressing oscillations in neural activity (see [39] for a detailed review). They are based (i) on population dynamics [46][15], (ii) on detailed single neuron descriptions [13], (iii) on simplified but computationally efficient models (e. g. Rulkov maps [47]), (iv) or on combinations thereof [14]. These approaches have their merits, but they come with limitations: (i) the parameters cannot be directly mapped to experimental measurable quantities (ii) it is not clear if the results scale to large networks of neurons. The approach that we presented here is a trade-off between biophysical realism and analytical tractability. We used the LIF model, which captures single-neuron dynamics to a sufficient degree, while at the same time allows computationally efficient simulations of large networks. We applied DFC that was originally proposed in the context of chaotic systems as a method to control unstable periodic orbits [5]. DFC has been also used to control dynamics in extended media [48] and has been applied in different contexts as well [49]. It was later used to control coherence [50] and to suppress synchronous activity in networks in which the neurons themselves act as oscillators [10–12]. Here, we did not use simplified population dynamics or phase oscillators. Instead, we used spiking neurons that fire irregularly and are nevertheless able to generate oscillations. We used a single proportional control term to provide a proof of principle of the method and to be able to delineate the control domains semi-analytically. Alternative approaches, e. g. linear PI/PID control [4] or non-linear control schemes [11] are also possible and may improve performance, however, their theoretical analysis is less straightforward. We also inserted realistic descriptions of synaptic dynamics and, therefore, were able to explicitly study their contribution to stability. This allowed us to design an appropriate control kernel, which resulted in increased control domains. In addition, by using a mean-field theory that explicitly incorporates the synaptic and neuronal response functions we could study their contribution in a systematic way. The neuronal response function enabled us to investigate the influence of external and recurrent inputs and to relate them to experimentally measurable quantities. Indeed, as we showed above, the statistics of the mean field for activity states with very similar firing rate profiles may be significantly different affecting stability. Therefore, feasible measurements of the population activity can be directly used to characterize the operating point of the network and to fine-tune the control parameters to achieve the desired results. Finally, the results presented here provide us with an understanding of the recent success of event-triggered control strategies. Event-triggered control can be placed between open-loop and continuous closed-loop control (e. g. DFC). Open-loop provides constant magnitude stimulation independent of the ongoing activity. Event-triggered approaches provide also constant magnitude stimulation, but only if a certain event occurs, for instance the power of beta oscillations crosses a certain threshold. Thus, the overall stimulation time and, therefore, the undesired stimulation side-effects are reduced. In DFC the stimulation side-effects are likely to be further reduced, because the stimulation amplitude is continuously adjusted to the ongoing activity. That is, no excessive stimulation is applied. This is particularly true for differential DFC, in which the stimulation amplitude vanishes over time. We used DFC, a relatively simple form of control that includes only a proportional gain term, because it is still possible to analytically study the stability of the closed-loop control system. More sophisticated control strategies could further increase the performance of the system. They come, however, at the price of increasing the number of control parameters that have to be estimated and of increasing complexity precluding a formal proof of stability. Our approach spans multiple levels of analysis of neuronal dynamics, enabling an understanding of how the control stimulus interacts with both low-level synaptic and high-level properties of the population activity to influence stability. At the same time the complexity of the controller is low enough to be of practical relevance. Thus, here we have provided a general conceptual framework for future studies that address both theoretical and practical aspects of closed-loop control in neuronal systems.
We simulate networks of N LIF neurons randomly connected with a probability of ϵ = 0. 1. Thus each neuron receives on average C = ϵN connections from other neurons in the network. For the purely inhibitory network we use N = NI and for the coupled excitatory-inhibitory case N = NE+NI. The subthreshold dynamics of a neuron i in the network is given by τ m d v i (t) d t = (v r e s t - v (t) ) + R m · I i, r e c (t) + R m · I i, e x t (t) (5) where Rm is membrane resistance, τm is the membrane time constant and vrest is the resting potential. The recurrent input term I i, r e c (t) = - ∑ j = 1 N J i j c i j ∑ k s (t - t j k - d i j) (6) describes the total synaptic current arriving at the soma due to presynaptic spikes. cij are elements of the binary connectivity matrix. Each presynaptic spike causes a stereotypical postsynaptic current s (t) modeled as an α-function [51] s (t) = t τ s e 1 - t t s H (t) (7) where τs is the synaptic time constant and H (t) the Heaviside function. The double sum in Eq 6 runs over all firing times t j k of all presynaptic neurons connected to neuron i. For all connections in the network we use the same synaptic coupling strength Jij = J/C, where C is the average in-degree and dij = d the transmission delay. The external input I i, e x t (t) = μ + σ τ m η i (t) (8) contains a mean term μ and a fluctuating term resulting from the Gaussian white noise ηi (t) that is uncorrelated from neuron to neuron with <ηi (t) > = 0 and <ηi (t) ηi (t′) > = δ (t−t′). In the stable asynchronous state the population average of the firing rate is constant in time, r (t) = r0. The mean recurrent input that each neuron receives is therefore also constant and given by μ r e c (t) = ⟨ J / C · ∑ c i j ∫ s (τ) ∑ k δ (t j k - τ - d) d τ ⟩ = J · r 0 · ∫ s (τ) d τ = μ r e c similarly the variance of the recurrent input is σ r e c 2 (t) = V a r [ J / C · ∑ c i j ∫ s (τ) ∑ k δ (t j k - τ - d) d τ ] = J 2 / C · r 0 · ∫ s 2 (τ) d τ = σ r e c 2 We study the stability of the asynchronous state following a linear perturbation approach [18]. A small oscillatory modulation of the stationary firing rate r (t) = r0+r1 e−λt with v1 ≪ 1 and λ = x+jω where ω is the modulation frequency leads to corresponding oscillation of the synaptic current I 1 = - J · r 1 · e · τ s (1 + λ · τ s) 2 e - λ d (9) The firing rate in response to an oscillatory input is given by r 1 = I 1 · r 0 σ (1 + λ τ m) ∂ U ∂ y (y t, λ) - ∂ U ∂ y (y r, λ) U (y t, λ) - U (y r, λ) (10) The function U is given in terms of combinations of hypergeometric functions U (y, λ) = e y 2 Γ (1 + λ · τ m 2) F 1 - λ · τ m 2,1 2, - y 2 + e y 2 Γ (λ · τ m 2) F 1 - λ · τ m 2,3 2, - y 2 In a recurrent network the modulation of the firing rate and the modulation of the synaptic input must be consistent. Combining Eqs (9) and (10) we get 1 = - J · r 0 · e · τ s e - λ d σ (1 + λ τ m) (1 + λ τ s) 2 ∂ U ∂ y (y t, λ) - ∂ U ∂ y (y r, λ) U (y t, λ) - U (y r, λ) which we write as 1 = J · R (λ) · S (λ) · e - λ d (11) where the terms R (λ) = 1 σ (1 + λ τ m) ∂ U ∂ y (y t, λ) - ∂ U ∂ y (y r, λ) U (y t, λ) - U (y r, λ) and S (λ) = e · τ s (1 + λ · τ s) 2 describe the neuronal and synaptic response functions respectively. The negative sign of J is absorbed in the phase of S (λ). The critical coupling values at which modes have marginal stability with frequency ωi can then simply be computed by J i = 1 R (ω i) · S (ω i) The smallest value Jcr = min{Ji} is the critical coupling at which the first complex pair of eigenvalues crosses the imaginary axis and the system becomes unstable. In the case of the inhibitory network for μ = 14 mV and σ = 6 mV we have Jcr ≈ 115 mV. In the simulations we used for the coupling between two neurons i and j, Jij = 0. 2mV thus the total coupling is J = C ⋅ Jij = 1000 ⋅ 0. 2 mV = 200 mV >Jcr (Fig 2a–2d). In the simulations we implement DFC by recording and stimulating all neurons in the network. The subthreshold dynamics of a neuron i with DFC is given by τ m d v i (t) d t = (v r e s t - v (t) ) + R m · I i, r e c (t) + R m · I i, e x t (t) + R m · I C (t) (12) where IC (t) is the control input. Note that IC (t) is identical for all neurons in the network given by I C (t) = K · m (t) ⋆ (v (t - d c) ) (13) where v (t) is the instantaneous population activity at time t and ⋆ denotes the convolution operation (f ⋆ g) (t) = ∫ - ∞ ∞ f (t - τ) g (τ) d τ. We used as control kernel m (t) a box function m (t) = H (t - a) - H (t - b) where H (t) is the Heaviside function H (t) = 0, t < 0 m s 1,0 ≤ t ≤ 1 m s Thus the control input IC (t) was updated in steps of 1ms.
Brain stimulation is being used to ease symptoms in several neurological disorders in cases where pharmacological treatment is not effective (anymore). The most common way for stimulation so far has been to apply a fixed, predetermined stimulus irrespective of the actual state of the brain or the condition of the patient. Recently, alternative strategies such as event-triggered stimulation protocols have attracted the interest of researchers. In these protocols the state of the affected brain area is continuously monitored, but the stimulus is only applied if certain criteria are met. Here we go one step further and present a truly closed-loop stimulation protocol. That is, a stimulus is being continuously provided and the magnitude of the stimulus depends, at any point in time, on the ongoing neural activity dynamics of the affected brain area. This results not only in suppression of the pathological activity, but also in a partial recovery of the transfer function of the activity dynamics. Thus, the ability of the lesioned brain area to carry out relevant computations is restored up to a point as well.
action potentials
medicine and health sciences
neural networks
population dynamics
membrane potential
electrophysiology
neuroscience
systems science
mathematics
algebra
computational neuroscience
population biology
computer and information sciences
system instability
animal cells
cell biology
linear algebra
single neuron function
neurons
physiology
biology and life sciences
cellular types
physical sciences
computational biology
eigenvalues
neurophysiology
Recovery of Dynamics and Function in Spiking Neural Networks with Closed-Loop Control