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. 2020 Aug 25;16(8):e1007790.
doi: 10.1371/journal.pcbi.1007790. eCollection 2020 Aug.

Firing rate homeostasis counteracts changes in stability of recurrent neural networks caused by synapse loss in Alzheimer's disease

Claudia Bachmann  1 , Tom Tetzlaff  1 , Renato Duarte  1 , Abigail Morrison  1   2
Affiliations

Firing rate homeostasis counteracts changes in stability of recurrent neural networks caused by synapse loss in Alzheimer's disease

Claudia Bachmann et al. PLoS Comput Biol. .

Abstract

The impairment of cognitive function in Alzheimer's disease is clearly correlated to synapse loss. However, the mechanisms underlying this correlation are only poorly understood. Here, we investigate how the loss of excitatory synapses in sparsely connected random networks of spiking excitatory and inhibitory neurons alters their dynamical characteristics. Beyond the effects on the activity statistics, we find that the loss of excitatory synapses on excitatory neurons reduces the network's sensitivity to small perturbations. This decrease in sensitivity can be considered as an indication of a reduction of computational capacity. A full recovery of the network's dynamical characteristics and sensitivity can be achieved by firing rate homeostasis, here implemented by an up-scaling of the remaining excitatory-excitatory synapses. Mean-field analysis reveals that the stability of the linearised network dynamics is, in good approximation, uniquely determined by the firing rate, and thereby explains why firing rate homeostasis preserves not only the firing rate but also the network's sensitivity to small perturbations.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Sketch of the network model of Alzheimer’s disease and homeostasis.
A) The network comprises two reciprocally and recurrently connected populations of excitatory (E) and inhibitory (I) integrate-and-fire neurons, excited by an external spiking input. Thickness of arrow indicates relative strength of the connection. In this study, Alzheimer’s disease is modeled by removing connections between excitatory neurons (loss of EE synapses) and upscaling of the remaining EE synapses to maintain the average firing rate (firing rate homeostasis). B–E) Sketch of EE connection density (number of arrows in upper panels), connection strength (thickness of arrows in upper panels) and resulting single-neuron spiking activity (lower panels). B) Intact network (without synapse loss). C) Synapse loss without homeostasis: removal of EE synapses and resulting reduction in firing rate. D) Synapse loss with unlimited homeostasis: removal of EE synapses and increase in strength of remaining EE synapses to maintain the average firing rate. Synaptic weights are allowed to grow without bounds. E) Synapse loss with limited homeostasis: removal of EE synapses and bounded increase in strength of remaining EE synapses. Here, synaptic weights cannot exceed 120% of their reference weight. The firing rate is therefore only partially recovered. For a complete description and parameter specification of the network model, see, Sec. Network model and, S1 and S2 Tables in the Supplementary Material.
Fig 2
Fig 2. Effect of different parameter configurations on the network’s firing rate.
The time and population-averaged firing rate is explored with respect to two parameters: the synaptic weight J and the loss of EE synapses. In A the EPSP amplitude is fixed (J ∈ {0.1, 1, 2, 3}mV, represented by a circle, square, triangle and diamond markers, respectively) and the firing rate ν is plotted against various degrees of synapse loss. In B the firing rate ν is plotted against the synaptic weight J for four different degrees of EE synapse loss (0%, 10%, 30%, 70%, with circle, square, triangle and diamond markers). The information of these two plot is combined in the contour plot C, which shows the dependence of the firing rate ν (colour coded) on synaptic reference weight J and the degree of synapse loss. Vertical rose lines correspond to the lines plotted in A, horizontal lines to the yellow lines in B. For visualization purposes, markers show only a subset of the data. All plotted data corresponds to the mean across 10 random realizations.
Fig 3
Fig 3. Effect of synapse loss and firing rate homeostasis on firing rate, synaptic weights and total synaptic contact area.
Dependence of the time and population averaged firing rate ν (A–C), synaptic weight JEE (D–F) and the relative total synaptic contact area (TSCA) of EE synapses (G–I) on the reference weight J and the degree of EE synapse loss in the absence of homeostatic compensation (left column), as well as with unlimited (middle column) and limited firing rate homeostasis (right column). Color-coded data represent mean across 10 random network realizations. Symbols mark parameter configurations shown in Fig 5.
Fig 4
Fig 4. Effect of synapse loss and firing rate homeostasis on spike train statistics.
Dependence of the coefficient of variation CV of inter-spike intervals (A–C) and the Fano factor FF of the population spike count (binsize b = 10ms; D–F) on the synaptic reference weight J and the degree of EE synapse loss in the absence of homeostatic compensation (left column), as well as with unlimited (middle column) and limited firing rate homeostasis (right column). Color-coded data represent mean across 10 random network realizations. Symbols mark parameter configurations shown in Fig 5.
Fig 5
Fig 5. Effect of synapse loss and firing rate homeostasis on spiking activity.
Spiking activity (dots mark time and sender of each spike) in an intact reference network (no synapse loss, JEE = 1.4mV; A), as well as in networks where 30% of the EE synapses are removed: B) no homeostasis (JEE = 1.4mV), C) unlimited homeostasis (JEE = 2.02mV), D) limited homeostasis (JEE = 1.68mV). In all panels, the synaptic-weight scale is set to J = 1.4mV. Examples depict parameter configurations marked by corresponding symbols in Figs 3 and 4 (cf. marker in lower right corner of each panel). The values for the parameters explored in these figures (synaptic weight J, EE synaptic weight JEE, total synaptic contact area TSCA, firing rate ν, coefficient of variation CV and Fano factor FF) are listed next to each plot. Regions below and above the gray horizontal line show spiking activity of a subset of 100 excitatory (E) and 25 inhibitory neurons (I), respectively.
Fig 6
Fig 6. Perturbation sensitivity.
Top: Example spiking activity (dots mark time and sender of each spike) of two identical networks (identical neuron parameters, connectivity, external input, initial conditions) with (black dots) and without perturbation (purple dots). The perturbation consists in delaying one external input spike at time t* = 400ms by δt* = 0.5ms. The vertical red line marks the time of the perturbation. Spikes of only 10% of all neurons are shown. Neurons below and above the horizontal gray line correspond to excitatory and inhibitory neurons, respectively. Bottom row: Perturbation sensitivity S(t) = 1 − |R(t)| obtained from the correlation coefficient R(t) of the low-pass filtered spike trains generated by the unperturbed and the perturbed network (black and purple dots in top panels; see, Sec. Synaptic contact area and characterization of network activity). A) Stable dynamics (J = 0.45mV, KEE = 100). B) Chaotic dynamics (J = 1.75mV, KEE = 100).
Fig 7
Fig 7. Effect of synapse loss and firing rate homeostasis on perturbation sensitivity.
Dependence of perturbation sensitivity S on the synaptic reference weight J and the degree of EE synapse loss in the absence of homeostatic compensation (A), as well as for unlimited (B) and limited firing rate homeostasis (C). Color-coded data represent mean across 10 random network realizations. Superimposed black and gray curves mark regions where the linearized network dynamics is stable (gray dashed; spectral radius ρ = …, 0.6, 0.8), about to become unstable (black; ρ = 1), and unstable (gray solid; ρ = 1.2, 1.4, …). Pink symbols mark parameter configurations shown in Figs 3 and 5.
Fig 8
Fig 8. Firing rate as predictor of linear stability and perturbation sensitivity.
Dependence of the linear stability quantified by the spectral radius ρ (A; theory) and the perturbation sensitivity S (B; simulation results) on the mean stationary firing rate νE of the excitatory neuron population in the absence of homeostasis (blue), as well as for unlimited (red) and limited firing rate homeostasis (yellow). Scatter plots depict data for various reference weights J ∈ {0, …, 3}mV and various degrees of synapse loss from 0% to 50%. Same data as in Figs 7, 12D–12F and 12M–12O.
Fig 9
Fig 9. Effect of EE synapse growth on firing rate and sensitivity.
Dependence of the firing rate (A, B) and the sensitivity (C, D) on the synaptic reference weight and the degree of EE synapse growth for an EE synapse loss of 30% (KEE = 70) (A, C) and no synapse loss (B, D). An EE synapse growth of zero means that all excitatory synapses have the same weight (JEE = JIE). For values larger than zero, synapses of excitatory neurons onto other excitatory neurons are stronger than onto inhibitory neurons. Black curve in A and C represent the firing rate and sensitivity of the reference network (no synapse loss, JEE = JIE). Color-coded data represents mean across 10 random network realizations.
Fig 10
Fig 10. Firing rate and not E/I balance as predictor of linear stability and perturbation sensitivity.
Dependence of perturbations sensitivity S on the E/I balance B (A) and on the mean stationary firing rate (B) for the limited homeostasis network (compare Fig 8, yellow), a network of full synaptic density (KEE = 100) but increased EE synapse growth (JEE ∈ {1 ⋅ JIE, …, 1.1 ⋅ JIE}; black) and for a network in which 30% EE synapse loss (KEE = 70) is compensated by EE synapse growth (JEE ∈ {1 ⋅ JIE, …, 1.5 ⋅ JIE}; red). Color-coded data represents mean across 10 random network realizations.).
Fig 11
Fig 11. The effect of local homeostasis on sensitivity.
Dependence of sensitivity to perturbations of networks with local homeostasis on different synaptic reference weight J ∈ {0.5mV, 1.2mV, 2.2mV} (A) and on the network’s population average firing rate ν. Red crosses: reference networks without synapse loss (KEE = 100); black dots: networks that underwent synapse loss but that regained the reference firing rate, assuming new synapses were created with a weight of Jlh = JEE; pink dots: as for black dots, but initial synapses created with a weight of Jlh = 6 ⋅ 10−4mV. (B) relationship between sensitivity and firing rate for all networks, regardless of whether the original firing rate was regained. Dots and circles as in (A); gray and pink triangles: as for black and pink dots, but for networks that could not restore the reference rate. All data points represent the mean across all realizations of a particular network weight J and a particular degree of synapse loss (0, 20, 40, 60, 80%).
Fig 12
Fig 12. Mean-field theory.
Dependence of the synaptic weight J^EE (A–C), the average firing rate νE of the excitatory population (D–F), the effective weight wEE of EE connections (G–I), the ratio wEEσE/J^EE (J–L), and the spectral radius ρ (M–O) on the synaptic weight J and the degree of synapse loss in the absence of homeostatic compensation (left column), as well as with unlimited (middle column) and limited firing rate homeostasis (right column). Superimposed black curves in (M–O) mark instability lines ρ = 1. Same parameters as in network simulations (see, S1 and S2 Tables in Supplementary Material).
Fig 13
Fig 13. Approximation of f(y)=ey2[1+erf(y)] by an exponential function.
A,B) Dependence of yrE (A) and yθE (B) on the synaptic reference weight J and the degree of synapse loss in the presence of unlimited firing rate homeostasis (mean-field theory). C) Graph of f(y)=ey2[1+erf(y)] (black) and exponential function AeBX (gray; A = 0.4, B = 2.5) fitted to f(y) in interval y ∈ [0.5, 1.5]. Same parameters as in network simulations (see, S1 and S2 Tables in Supplementary Material).
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Grants and funding

This project was funded by the Helmholtz Association Initiative and Networking Fund (project no. SO-092 [Advanced Computing Architectures], and Helmholtz Portfolio Theme "Supercomputing and Modeling for the Human Brain"), the European Union’s Horizon 2020 Framework Programme for Research and Innovation under Specific Grant Agreement No. 720270 and 785907 (Human Brain Project SGA1 and SGA2), and the German Research Foundation (DFG; grant DI 1721/3-1 [KFO219-TP9]). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The authors also gratefully acknowledge the computing time granted by the JARA-HPC Vergabegremium on the supercomputer JURECA at Forschungszentrum Jülich.