Fig 1.
Formation of memory engrams in a neuronal network with homeostatic structural plasticity.
(A) In a classical conditioning scenario, an unconditioned stimulus (US) was represented by a group of neurons that were connected to a readout neuron (yellow) via static synapses. The readout neuron spiked whenever neuronal activity representing the US was above a certain threshold (top). During a conditioning protocol, two other groups of neurons (CS1 and CS2) were chosen to represent two different conditioned stimuli (bottom). (B) During stimulation, the external input to a specific group of neurons was increased. The color marks indicate when each specific group was being stimulated. During the “encoding” phase, CS1 was always stimulated together with US, and CS2 was always stimulated alone. The time axis matches that of panel C (top). (C) Firing rate (top) and spike train (bottom) of the readout neuron. Before the paired stimulation (“baseline”), the readout neuron responded strongly only upon direct stimulation of the neuronal ensemble corresponding to the US. After the paired stimulation (“retrieval”), however, a presentation of C1 alone triggered a strong response of the readout neuron. This was not the case for a presentation of C2 alone. (D) Coarse grained connectivity matrix. Neurons are divided into 10 groups of 100 neurons each, shown is the average connectivity within each group. After encoding, the connectivity matrix indicates that engrams were formed, and we found enhanced connectivity within all three ensembles as a consequence of repeated stimulation. Bidirectional inter-connectivity across different engrams, however, is only observed for the pair C1-US that experienced paired stimulation. (E) Average connectivity as a function of time. The connectivity dynamics shows that engram identity was strengthened with each stimulus presentation, and that engrams decayed during unspecific external stimulation.
Fig 2.
Silent memory based on structural engrams.
(A–C) The ongoing activity of neurons belonging to the engram E1 can hardly be distinguished from the activity of the rest of the network. (A) Raster plot showing the spontaneous activity of 50 neurons randomly selected from E1, 100 neurons randomly selected from E2 but not belonging to E1, and 50 neurons randomly selected from the pool I of inhibitory neurons. (B) Overlap of ongoing network activity with the learned engram E1 (, orange), and separately for 10 different random ensembles x disjoint with E1 (mx, purple). (C) Cumulative distribution of mμ shown in (B). (D–F) The activity evoked upon stimulation of E1 is higher, if the within-engram connectivity is large enough (
) as a consequence of learning. (D) Same as A for evoked activity, the stimulation starts at t = 1 s. The neurons belonging to engram E1 are stimulated before (top,
) and after (bottom,
) engram encoding. (E) Overlap with the learned engram (
, orange) and with random ensembles (mx, purple) during specific stimulation of engram E1. (F) Population rate of all excitatory neurons during stimulation of E1 before (black) and after (orange) engram encoding. (E, F) Solid line and shading depict mean and standard deviation across 10 independent simulation runs, respectively. In all panels, the bin size for calculating overlaps is 10 ms, and the bin size for calculating population rates is 100 ms.
Table 1.
List of symbols.
Fig 3.
Evoked activity depended on the strength of a memory.
(A) Starting from a random network grown under the influence of unstructured stimulation (black dot), we repeatedly stimulated the same ensemble of excitatory neurons E1 to eventually form an engram. Multiple stimulation cycles increased the recurrent connectivity within the engram. (B) Population activity of all excitatory neurons upon stimulation of E1, for different levels of engram connectivity . Crosses depict the population rate observed in a simulation. Colors indicate engram connectivity
, matching the colors used in panels (A) and (D). The grey line outlines the expectation from a simple mean-field theory. (C) Firing rate of E1 engram neurons upon stimulation of 50% its neurons. Shown is the mean firing rate of the stimulated engram neurons (E1S, orange), the mean firing rate of the non-stimulated engram neurons (E1N, grey), and the mean firing rate of excitatory neurons not belonging to the engram (E2, blue). Solid line and shading depict mean and standard deviation for 10 independent simulation runs, respectively. (D) Time-averaged overlap
, for different fractions of E1 being stimulated. The recurrent nature of memory engrams enabled them to perform pattern completion. The degree of pattern completion depended monotonically on engram strength. (E) Two engrams (orange and green) were encoded in a network. Both engrams had a different strength with regard to their within-engram connectivity (green stronger than orange). A simple readout neuron received input from a random sample comprising 9% of all excitatory and 9% of all inhibitory neurons in the network. (F) Raster plot for the activity of 10 different readout neurons during the stimulation of learned engrams and random ensembles, respectively. Readout neurons were active when an encoded engram was stimulated (orange and green), and they generally responded with higher firing rates for stronger engrams (green). The activity of a readout neuron was low in absence of a stimulus (white), or upon stimulation of a random ensemble of neurons (purple and blue).
Fig 4.
Hebbian properties emerge through the interaction of selective input and homeostatic control.
(A) The activity of the neuronal network was subject to homeostatic control. For increased external input, it transiently responded with a higher firing rate. With a certain delay, the rate was down-regulated to the imposed set-point. When the stimulus was turned off, the network transiently responded with a lower firing rate, which was eventually up-regulated to the set-point again. The activity was generally characterized by irregular and asynchronous spike trains. (B) It was assumed that the intracellular calcium concentration followed the spiking dynamics, according to a first-order low-pass characteristic. Dots correspond to numerical simulations of the system, and solid lines reflect theoretical predictions from a mean-field model of dynamic network remodeling. (C) Dendritic elements (building blocks of synapses) were generated until an in-degree of Kin = 1000 was reached. This number slightly decreased during specific stimulation, but then recovered after the stimulus was removed. (D) Synaptic connectivity closely followed the dynamics of dendritic elements until the recovery phase, when the recurrent connectivity within the stimulated group E1 overshooted. (A–D) Black vertical lines indicate beginning and end of the stimulation. (E, F) Phase space representation of the activity. The purple lines are projections of the full, high-dimensional dynamics to different two-dimensional subspaces: (E) within-engram connectivity vs. across-ensemble connectivity and (F) within-engram connectivity vs. engram calcium trace. The dynamic flow was represented by the gray arrows. The steady state of the plastic network was characterized by a line attractor (thick gray line), defined by a fixed total in-degree and out-degree. The ensemble of stimulated neurons formed a stable engram, and the strength of the engram was encoded by its position on the line attractor. (G) The effective “instantaneous” learning rule for the expected connectivity between a pair of neurons is homeostatic in nature. It could also be viewed as an “inverse covariance rule” with baseline at the set-point of the homeostatic controller. The emerging Hebbian properties results from the more long-term combinatorial properties of rewiring across the whole network.
Fig 5.
Noisy spiking induces fluctuations that lead to memory decay.
(A) The gray histogram shows the distribution of calcium levels for a single neuron across 5000 s of simulation. The yellow lines resulted from modeling the spike train as a Poisson process (dotted line) or a Gamma process (solid line), respectively. (B) The rate of creation or deletion of synaptic elements depends on the difference between the actual firing rate from the target rate (set-point), for different levels of spiking noise. The negative gain (slope) of the homeostatic controller in presence of noise is transformed into two separate processes of creation and deletion of synaptic elements. In the presence of noise (grey lines, lighter colors correspond to stronger noise), even when the firing rate is on target, residual fluctuations of the calcium signal induced a continuous rewiring of the network, corresponding to a diffusion process. (C) If noisy spiking and the associated diffusion was included in the model, our mean-field theory matched the simulation results very well. This concerns the initial decay, the overshoot and subsequent slow decay. Vertical lines indicate the beginning and the end of the stimulation. (D) Change in connectivity during the decay period, for different values of the calcium time constant (different shades of red, from light to dark τCa = 1, 2, 4, 8, 16, 32 s, ρ = 8 Hz) and the target rate (different shades of purple, from light to dark ρ = 2, 4, 8, 16, 32 Hz, τCa = 10 s). We generally observed exponential relaxation as a consequence of a constant rewiring rate. (E) Time constant of the diffusive decay as a function of the calcium time constant and the target rate. Lines show our predictions from theory, and dots represent the values extracted from numerical simulations of plastic networks. The decay time τdecay increases with . The memory was generally more stable for small target rates ν, but collapsed for very small rates. This indicates an optimum for low firing rates, at about 3 Hz. (F, G) Same phase diagrams as shown in Fig 4E and 4F, but taking noise into consideration. (F) The spiking noise compromises the stability of the line attractor, which turns into a slow manifold. (G) The relaxation to the high-entropy connectivity configuration during the decay phase is indeed confined to a constant firing rate manifold.
Fig 6.
Linear stability of a network with homeostatic structural plasticity.
(A) For a wide parameter regime, the structural evolution of the network has a single fixed point, which is also stable. Three typical types of homeostatic growth responses are depicted for this configuration: non-oscillatory (left), weakly oscillatory (middle), and strongly oscillatory (right) network remodeling. (B) All eigenvalues of the linearized system have a negative real part, for all values of the growth parameters of dendritic (axonal) elements βd and calcium τCa. In the case of fast synaptic elements (small βd) or slow calcium (large τCa), the system exhibits oscillatory responses. Shown are real parts and imaginary parts of the two “most unstable” eigenvalues, for different values of βd (left column, τCa = 10 s) and τCa (right column, βd = 2). (C) Phase diagram of the linear response. The black region below the red line indicates non-oscillatory responses, which corresponds to the configuration τCa ≤ 3βd s. Dots indicate the parameter configurations shown in panel (A), with matching colors.
Fig 7.
Non-linear stability of a network with homeostatic structural plasticity.
(A) The engram E1 is stimulated with a very strong external input. As the homeostatic response triggers excessive pruning of recurrent connections, the population E1 is completely silenced after the stimulus is turned off. This, in turn, initiates a strong compensatory overshoot of connectivity and consecutive runaway population activity. The dots with corresponding color show the results of a plastic network simulation, and the solid lines indicate the corresponding predictions from our theory. The theoretical instantaneous firing rate is clipped at 100 Hz. (B) The network settles in a limit cycle of connectivity dynamics. The hysteresis-like behavior is caused by the faster growth of within-engram connectivity as compared to connectivity from the non-engram ensemble
. During the initial phase of the cycle, the increase of
has no effect on the activity of population E1 yet, as its neurons are not active. Only when the input from population E2 through
gets large enough, the rate
becomes non-zero and rises to very high values quickly due to already large recurrent
connectivity. (C) The calcium signal ϕ adds an additional delay to the cycle. (D) This leads to smoother trajectories when scattering calcium concentration against connectivity. (E) Connectivity within the stimulated group
plotted against input connectivity from the non-engram population
. The black line shows configurations with constant in-degree, of which the black dot represents the most entropic one. The red dot corresponds to critical connectivity, beyond which the limit cycle behavior is triggered. The limit cycle transients in connectivity space are orthogonal to the line attractor, indicating that the total in-degree is oscillating and no homeostatic equilibrium can be established.