Local (a) and distributed (b) network architectures. Here, nodes a, b, i, s, and P denote two excitatory populations, fast and slow inhibitory populations, and pool population, respectively. Red edges represent excitatory couplings and blue edges represent inhibitory couplings. In the local network, the self coupling is stronger than inter-population coupling. Also, in the distributed network, inter-area connection is weaker than intra-area connection.
(a) Network response to direct stimulation of a selected excitatory population across different stimulus frequencies and amplitudes. Inhibitory populations also receive reduced direct stimulation (25% that of the excitatory population). Green areas (“WTA” for winner takes all) indicate frequencies and amplitudes of stimulation evoking normal working memory behavior in the network with the directly stimulated excitatory population exhibiting persistent increased firing rates above baseline. Dark blue areas (“BL” for baseline) of the diagram indicate frequencies and amplitudes not producing working memory behavior with population activity returning to baseline states following termination of input. Working memory behavior is evoked by any frequency and continuous input with amplitude between approximately 20 a.u. and 160 a.u. Above 160 a.u., working memory response exhibits a strong frequency dependence on the stimulus. (b) Top: Peristimulus time histograms (PSTH) showing the typical working memory behavior corresponding to the green area of (a). a and b represent excitatory populations, i fast inhibitory population, s slow inhibitory population, and p pool population (see Figure 1 ). A 50 Hz, 100 a.u. stimulus is given directly to one of the excitatory populations of the network (population “a” red trace) for 100 ms, beginning at T = 100 ms. The other local population of the network (population “b” blue trace) receives indirect stimulation via inter-population excitatory connections between “a” and “b.” Following the end of the stimulus period (at T = 200 ms), population “a” maintains persistent elevated firing, while population “b” returns to its baseline state. Below: array plot showing the activity as in the PSTH including all populations in the network. (c) PSTH (top) and array plot (bottom) showing a second type of stable working memory dynamics (persistent oscillatory WTA) emerging in the system from a Hopf bifurcation that occurs at particular values of I–E connection strength. Oscillatory WTA and persistent continuous WTA working memory behavior coexist over a range of the I-E connection strength. (d) Network response to direct stimulation of multiple populations. Population “b” receives a fixed stimulus (50 Hz, 500 a.u.). The response of the network is shown for population “a” receiving different inputs. Green (“WTA”) means population “b” wins, while cyan (“WTA switched”) means that population “a” wins. The system exhibits WTA working memory behavior with significant frequency dependence on the input. (e) Termination of working memory activity by a second stimulation. Top: Phase diagram showing amplitude and frequency values of the input resulting in the system transitioning to baseline (dark blue areas, “BL”). Green areas (“WTA”) indicate areas in which working memory behavior persists. Bottom: PSTH showing activation and successful termination of working memory behavior. (f) Diagrams indicating the mechanism for working memory termination. Successful termination of persistent activation by stimulation occurs if, at the end of the stimulus period, fast and slow inhibition (traces i and s) have built to a sufficient magnitude to drive the excitation (a) to the basin of attraction of its baseline state. In the top figure, the period of the stimulus is such that at the time of input termination (T = 1100 ms), s = 2 a.u. and i = 10 a.u. In the bottom figure, stimulus frequency is slightly reduced relative to the top figure (falling in the green area in E), and at the time of termination s = 2 a.u. and i = 6 a.u., which is below threshold for terminating the persistent activation.
Rapid transition to other memory states as a result of new network input. (a) Transitioning from persistent continuous WTA dynamics. For sufficiently low amplitude input (approximately 25 to 50 a.u.), the system remains in its present WM state and does not transition (green area, “WTA”). If the amplitude of stimulation is sufficiently high, the network transitions to persistent activation of the new WM state (cyan area, “WTA switched”). For sufficiently high stimulation (approximately greater than 150 to 250 a.u.), the system transitions to the new memory state but exhibits significant frequency dependence. The system successfully switches memory states at specific frequency bands, while for interim frequency bands all persistent activation terminates and all populations of the network return to their baseline state (dark blue area, “BL”). (b) Top: PSTH showing switching activation of the local network from population “a” (redtrace), which was the initial WM state activated by the memorandum stimulus, to activation of population “b” (blue trace) by a subsequent stimulus delivered directly to “b,” with concomitant termination of the persistent activation of “a” to baseline. Bottom: Array plot showing the activity in the PSTH including all populations in the network. (c) Transitioning from persistent oscillatory WTA dynamics. The network can successfully transition to a new working memory state (light blue area, “OWTA switched”) from a currently active one for stimulation with sufficiently high amplitudes (e.g., greater than approximately 10a.u.) but with a complex dependence on the frequency and amplitude of the new stimulus. For sufficiently high amplitudes (approximately greater than 300a.u.), stimulation drives all populations of the network to baseline (dark blue). (d) Top: PSTH showing the successful transition from one oscillatory WTA working memory state (red trace) to another oscillatory WTA working memory state (blue trace evoked by a second stimulus). Bottom: Array plot depicting the activity in the PSTH diagram above for all network populations.
Phase space showing the states of the local network as a function of critical parameters P, the inter-population connection density, and gie , the inhibitory to excitatory connection strength for various stimulation configurations. (a) Phase space for direct periodic input to a single population with amplitude 500a.u. and frequency 50 Hz. (b) Phase space for direct periodic input to a single population with amplitude 100 a.u. and frequency 50 Hz. Note the occurrence of oscillatory WTA working memory, which occurs for larger values of I-E coupling strength at normal levels of inter-population connection density. (c)Phase space for direct periodic input to multiple populations with different amplitudes (500 a.u. vs 100 a.u.). (d) Phase space for direct periodic input to a single population. Input is reduced to inhibitory populations (inhibitory population input 5% of excitatory population input), equivalent to a reduction in the number of inhibitory interneurons. Note that for all cases the essential states and transition boundaries between them remain relatively consistent, with the exception of the emergence of oscillatory WTA states ((b) and (d)) for larger values of I–E coupling (e.g., at gie > 35) and at modest stimulation amplitudes (100 a.u.).
(a) Bifurcation diagram of local network as a function of changing I–E coupling strength, at excitatory connection density p = 0.9 (10% of inter-population connectivity relative to self-connectivity). y-axis is firing rate of the excitatory population a. Baseline is stable for all values of the parameter gie (fixed line near zero firing rate not shown). At relatively high I–E connection strength (e.g., gie > 45), only the baseline is stable. At gie = 35, a Hopf bifurcation occurs leading to an unstable branch with stable oscillatory WTA states (OWTA). The oscillatory WTA branch is present over a range of gie from approximately 34 to at least 44. Continued decrease of gie leads to bifurcations producing WTA working memory states and bistability with elevated firing rates. WTA behavior persists with monotonically increasing stable firing rates for decreasing gie . At gie = 15 down to gie = 2.5, the system exhibits tri-stability with the emergence of a seizure branch (all populations asynchronously active at a fixed frequency) arising from a saddle node bifurcation occurring at gie = 2.5. Below gie = 2.5, only the baseline state and the seizure state are stable. (b) Bifurcation diagram at p = 0.5 corresponding to much greater inter-population connection density. Baseline remains the only stable state above gie = 35. At gie = 35, a Hopf bifurcation occurs leading to unstable branch of OWTA solutions with intermittent stable points. As gie decreases further, there is bistability with WTA and baseline states. At approximately gie = 25, a second Hopf bifurcation occurs leading to a WTA branch terminating at gie = 22, an unstable OWTA branch and the emergence of a seizure branch (asynchronous seizure state, “all-up”). Below gie = 22, only the baseline and seizure states are stable. In both cases ((a) and (b)), normal working memory behavior co-exists with seizure states. (c) Two-parameter diagram. The red curve depicts the fold curve representing the loss of the all-up state. All points to the left of the red curve allow for stable seizure states. The two black curves are the curves of Hopf bifurcations. At high values of p, to the right of HBw, there will be OWTA dynamics.
Response to stimulation of the network in the seizure state. The network exhibits tri-stability over the parameter range considered (as indicated in the bifurcation diagram, Figure 5 ) with the seizure state coexisting with a stable persistent WTA state and the baseline state. (a) Response of the network in the asynchronous seizure state (red, “all-up”) to stimulation given directly to a selected excitatory population and to inhibitory populations with reduced amplitude (25% of excitatory population stimulation). For input of sufficient amplitude, the network transitions to a working memory state (green, “WTA”). The ability of the network to make the transition exhibits dependence on the frequency of the stimulus at intermediate amplitudes (approximately 400 a.u. to 1000 a.u.). For a single input, the system does not transition directly to the baseline at any reasonable amplitude of stimulation. (b) A subsequent stimulation of sufficient amplitude to the system, which has been stimulated from the asynchronous seizure state to the working memory state, can transition the network to the baseline state. Top: PSTH showing (1) at T = 100 ms, stimulation of the network evoking seizure activity (all populations simultaneously persistently active), (2) a second stimulation at T = 1000 ms given directly to population “a” transitions the network to persistent working memory activation (blue trace corresponding to population “b” transitions to baseline, red trace corresponding to population “a” maintains selective persistent activation), (3) a third strong stimulation given to population “a” results in the termination of all persistent activity with all populations in the baseline state. Bottom: Array plot showing the activity in the PSTH for all network populations. (c) The transition of the seizure state to the working memory state succeeds provided the amplitude of the input is such that it enables slow inhibition to build above a threshold and fast inhibition to be sufficiently high so that all populations except the directly stimulated excitatory population are strongly suppressed during the stimulation period and remain at the baseline when the stimulation terminates (left).
Working memory dynamics in a distributed cortical network. (a) Stimulation to a single excitatory population results in the persistent activation of a distributed working memory network, which includes the directly excited population as well as populations in other areas to which the stimulated population reciprocally connects. The activated network exhibits persistent WTA state. (b) Stimulation to a single excitatory population evoking persistent activation of a distributed working memory network with the selected network populations exhibiting oscillatory WTA dynamics. (c) Phase diagram showing the states activated in the network from direct inputs given to multiple populations in network 1. Input to population “1b” in network 1 is fixed at 50 Hz, and 500 a.u. while the frequency and amplitude of the stimulation given to population “1a” of network 1 is varied. When the stimulus amplitude to population “1a” sufficiently lower than that given to population “1b,” the system persistently activates the network associated with population “1b” (i.e., green area (“WTA”) in diagram corresponding to activation of population “1b” in network 1 and population “2b” in network 2). When the stimulus amplitude given to “1a” is sufficiently greater than that given to “1b,” the network associated with population “1a” is persistent activated (blue area (“WTA switched”) corresponding to persistent activation of population “1a” in network 1 and population “2a” in network 2). When stimulation amplitudes are of similar amplitude, the particular network activated shows a significant frequency dependence of the stimulation. (d) Termination of the persistent WTA state from a second synchronizing stimulation given to all populations.
Phase space showing the states of the distributed network as a function of critical parameters p = inter-population connection density and d = inter-area connection density for various levels of I-E connection strengths. (a) Top: Phase space for direct periodic input to a single population in network 1 (left) with amplitude 500 a.u. and frequency 50 Hz within normal range of I-E connection strength (gie = 25). Change in p occurs in both networks. Bottom: Enlarged view of phase diagrams above in region showing transition boundaries between focal seizures, generalized seizures, and normal working memory dynamics. (b) Phase diagram for direct single periodic input to a single population in network 1 (left) with increased I–E coupling (gie = 35). The system exhibits oscillatory WTA working memory behavior for normal levels of p (i.e., >0.85) essentially independent of d values. As inter-population connection density increases (p < 0.85), the system bifurcates and exhibits beta/gamma oscillatory synchronous generalized seizures. (c) For decreased values of I–E coupling (gie = 15), the network exhibits distributed working memory behavior independent of d for normal values of p (i.e., greater than 0.9). For values of d greater than 0.15, the system bifurcates and exhibits asynchronous seizure behavior for values of p between 0.8 and 0.9. For inter-area connection densities smaller than approximately 0.15, as p decreases below normal levels the system transitions to asynchronous generalized seizures.
Behavior of the distributed network from Figure 8(a) showing the transition from focal seizures exhibiting oscillatory dynamics with increasing frequency to generalized seizures with increasing inter-area connection density d for local inter-population connectivity densities p = 0.61 in networks 1 and 2 (see vertical white arrow in bottom of Figure 8(a) ). (a) Direct stimulation of the excitatory population 1a results in that population being excited to a stable fixed point attractor and exhibiting persistent working memory type activation. This serves as an ongoing persistent source of excitation to the populations in network 2 after the stimulus has terminated. For the particular value of d, the amplitude to which the populations of network 2 are excited is insufficient to be maintained due to fast and slow inhibition rising above the threshold to drive the activity to baseline prior to the end of the stimulus period. The inhibition must decay sufficiently so that slow excitation resulting from the persistent excitatory input from the population in network 1 may build to a level that the populations of network 2 may activate to an elevated level again. This, in turn, causes increased fast inhibition and building slow inhibition in network 2, which subsequently drives the activity to baseline again. The particular level of the stable elevated activation in network 2, and the time constant for the slow inhibition along with the delay between the input from network 1 to network 2, thus determine the rate of the cycle which is in the theta range (white star in bottom of Figure 8(a) ). (b) With increased inter-area connection density, the populations of network 2 receive increased input from the persistently active population of network 1, resulting in an increased amplitude of firing of the populations in network 2. This results in the populations in network 2 rising above the level necessary to fire more rapidly than in (a) and producing synchronous seizure activity with an increased oscillation rate in the high beta/low gamma range (white circle in bottom of Figure 8(a) ). (c) As d increases sufficiently, for stimulation of the appropriate duration, network 2 populations are able to drive activity in the inhibitory populations of network 1 so that they may terminate the activity of network 1 to baseline at the end of the stimulation period. The sustained activation of network 2 then drives the activity in network 1, which in turn further stimulates the activity of the inhibitory populations in network 2, raising it beyond threshold so that network 2 activity terminates to baseline, concomitant with the rise of the populations of network 1. This cycle repeats with the activity of the populations (e and i) 180° out of phase and activating and shutting each other down in an alternating manner with a oscillatory frequency in the high beta/low gamma range (white triangle in bottom of Figure 8(a) ).
Possible focal and generalized seizure dynamics occurring nested within regions of the phase space (Figure 8 ) corresponding to a range of dynamics similar to those observed in seizures in the human cortex. Different seizure dynamics can be obtained by changing the critical parameters d, p, and gie for each network either asymmetrically ((a)–(c)) or symmetrically ((d)–(g)). (a) Generalized seizure with high frequency ripples in the populations of network 1 and high frequency oscillations exhibited by the populations of network 2. (b) Spike and wave activity in network 1 populations with periodic bursting in populations of network 2. (c) Delayed onset of ripples in network 1 and regular bursting in network 2. (d) Nested oscillations with synchronous gamma oscillations modulated at a theta frequency. (e) Delayed onset of synchronous theta oscillations. (f) High frequency (gamma) oscillations followed by periodic bursting in the theta band. (g) Sudden onset of high frequency synchronous oscillations in all populations. Networks in different areas are 180° out of phase.
Response to stimulation of the distributed network in a generalized seizure state exhibiting synchronized gamma oscillations (orange area, “sync”). (a) Stimulation of all populations transitions the system to baseline state (dark blue area, “BL”) if input amplitude is sufficiently large. (b) Stimulation of a local network: Termination of the seizure can occur but is strongly frequency dependent occurring for very specific inputs. (c) The mechanism of frequency dependent seizure termination in (b). Driving the network from a single local area results in synchronization of the activity at the end of stimulation for only very specific frequencies and amplitudes of input such that the mutual inhibition drives activity to its baseline state. In the first diagram (left), the frequency of the stimulus and its duration is such that when it terminates, the activities of the excitatory populations of both networks are synchronized at the same amplitude, as is the activity of inhibitory populations whose activity is of sufficient magnitude to shut down the excitatory activity to baseline. If the frequency of stimulation is slightly greater (center) or slightly less (right), then the activity of the excitatory and inhibitory populations is not in phase and of similar magnitude so that one or the other network remains active and enables the anti-phase seizure cycle to continue once stimulation has terminated. (d) Distributed input to multiple populations is more efficacious in terminating the seizure and transitioning the system to the baseline state than a stimulation of a local network, even though the overall level of stimulation is the same. In contrast to multiple populations within a single network, input to all networks is able to synchronize excitatory and inhibitory population activity over a broader range of stimulus parameters and thus terminates seizures for a broader range of frequencies and amplitudes of the input. (e) Response of the network in the seizure state to a stimulus directly given to a single population. For input of sufficient amplitude, the network transitions to a working memory state (green area, “WTA”). This transition exhibits dependence on the frequency of the stimulus at intermediate amplitude. Subsequent stimulation of the network can result in driving the network to its baseline state as in Figure 6(b) .
Response to stimulation of distributed network in focal seizure states. Stimulation may be given to the area exhibiting seizure dynamics or to another with projections to and from the pathological network. (a) System exhibiting focal seizure with synchronous beta/gamma oscillations with different peak amplitudes (orange area, “beta”) in network 1 and a single population in elevated oscillatory WTA (dark green area, “EOWTA”) in network 2. Stimulation directly to a single population of the network not exhibiting the seizure activity (population 2b in this case) results in the seizure activity persisting with peak amplitudes swapped (dark orange area, “beta switched”) in network 1, and changing the persistent activation to another population (blue area, “EOWTA switched”) in network 2. (b) Stimulation of all populations of the network exhibiting the focal seizure does not terminate the seizure behavior for any amplitude or frequency. If input is of sufficient amplitude for a given frequency, the system may actually transition to a generalized seizure state with the connected network exhibiting persistent activation of a single population transitioning to a state in which all populations are active and exhibiting synchronous oscillations (overlapping orange areas in left and right diagrams). (c) Distributed input to selected populations in networks 1 and 2 respond similarly to that of a single input to the pathological network 1, except for input with very low frequency and high amplitude, which can terminate the seizure. (d) Stimulation of the non-seizure network with projections to the network exhibiting seizure dynamics results in transitioning the system to baseline and termination of the seizure over a wide range of stimulation frequencies and amplitudes. Termination of the seizures of this type can thus be seen to be most effective, therefore, when stimulating a network exhibiting normal dynamics, which is reciprocally connected to the area exhibiting seizures.
Response to stimulation of distributed network in focal seizures states. System exhibiting focal seizures with synchronous theta oscillations. (a)Stimulation of a single population in the network exhibiting normal working memory maintains seizure activity in other network at periodic frequencies and over specific amplitudes, but is also capable of terminating the seizure to baseline for a wide range of particular frequencies and amplitudes. (b) Distributed stimulation of the system results in more efficacious termination of the seizure state. Stimulation of both the seizing and connected non-seizing networks results in termination of the seizure for all frequencies and amplitudes above approximately 30 Hz. Note that stimulation of all populations (not shown) results in transitioning the system to baseline for essentially all frequencies and amplitudes of the input.
Article metrics loading...
Full text loading...